Variable structure systems towards the 21st century

Lecture Notesin Control and Information Sciences 274

Editors: M. Thoma · M. Morari

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Xinghuo Yu, Jian-Xin Xu (Eds)

Variable Structure Systems:Towards the 21st CenturyWith 116 Figures

1 3

Series Advisory BoardA. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic ·A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

EditorsXinghuo Yu, Associate Professor Jian-Xin Xu, Associate ProfessorCentral Queensland University National University of SingaporeFaculty for Informatics Department of Electrical Engineeringand Communication SingaporeRockhamptonAustralia

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Preface

The book is a collection of contributions concerning the theories, applicationsand perspectives of Variable Structure Systems (VSS).

Variable Structure Systems have been a major control design methodologyfor many decades. The term Variable Structure Systems was introduced inthe late 1950’s, and the fundamental concepts were developed for its mainbranch Sliding Mode Control by Russian researchers Emelyanov and Utkin.The 20th Century has seen the formation and consolidation of VSS theoryand its applications. It has also seen an emerging trend of cross-fertilizationand integration of VSS with other control and non-control techniques such asfeedback linearization, flatness, passivity based control, adaptive and learningcontrol, system identification, pulse width modulation, H∞ geometric andalgebraic methods, artificial intelligence, modeling and optimization, neuralnetworks, fuzzy logic, to name just a few. This trend will continue and flourishin the new millennium.

To reflect these major developments in the 20th Century, this book in-cludes 16 specially invited contributions from well-known experts in VSStheory and applications, covering a wide range of topics.

The first chapter, “First Stage of VSS: People and Events” written byVadim Utkin, the founder of VSS, oversees and documents the historicaldevelopments of VSS in the 20th Century, including many interesting eventsnot known to the West until now.

The second chapter, “An Integrated Learning Variable Structure ControlMethod” written by Jian-Xin Xu, addresses an important issue regardingcontrol integration between variable structure control and learning control.

The third chapter, “Discrete-time Variable Structure Control” co-authoredby Katsuhisa Furata and Yaodong Pan, describes the design and analysis ofdiscrete-time variable structure control.

The fourth chapter, “Higher-Order Sliding Modes for the Output-FeedbackControl of Nonlinear Uncertain Systems” written by Giorgio Bartolini, ArieLevant, Alessandro Pisano and Elio Usai, discusses the properties of higher-order sliding mode and its application to output feedback control tasks.

The fifth chapter, “Variable Structure Systems with Terminal SlidingModes” written by Xinghou Yu and Man Zhihong, suggests the utility ofa terminal attractor to improve the convergence performance of VSS in slid-ing modes.

The sixth chapter, “Adaptive Backstepping Control” written by Ali Ja-fari Koshkouei, Russell Mills, Allan Zinober, applies the backstepping designmethod to sliding mode control and combines adaptive mechanism.

The seventh chapter, “Sliding Mode Compensation, Estimation and Op-timization Methods in Automotive Control” written by Ibrahim Haskara,

ii

Cem Hatipoglu and Umit Ozguner, which demonstrates the effectiveness ofthe sliding mode control and estimation approaches in solving automotivecontrol problems.

The eighth chapter, “On Quasi-optimal Variable Structure Control Ap-proaches” written by Jian-Xin Xu and Jin Zhang, explores the possibility ofcombining sliding mode control with nonlinear optimal control methods.

The ninth chapter, “Robust Control of Infinite-Dimensional Systems viaSliding Modes” written by Yuri Orlov, deals with an important area: how toconstruct a sliding mode controller for infinite dimensional systems.

The tenth chapter, “Sliding Modes Applications in Power Electronics andElectrical Drives” written by Asif Sabanovic, Karel Jezernik and NadiraSabanovic, summarizes sliding mode applications in power electronics andelectrical drives.

The eleventh chapter, “On the Development and Application of SlidingMode Observers” written by Chris Edwards, Sarah Spurgeon and Chee PinTan, illustrates recent developments and applications of sliding mode ob-servers.

The twelfth chapter, “Multivariable Output-Feedback Sliding Mode Con-trol” written by Liu Hsu, Jose Paulo Vilela Soares da Cunha, Ramon R.Costa and Fernando Lizarralde, examines output feedback issues associatedwith sliding mode control.

The thirteen chapter, “Sliding Modes, Differential Flatness and IntegralReconstructors” written by Herbertt Sira-Ramirez and Victor Hernandez,studies the connections between sliding mode and flatness/integral recon-struction.

The fourteenth chapter, “On Robust VSS Nonlinear ServomechanismProblem” is written by Vadim Utkin, B. Castillo-Toledo, A. Loukianov andO. Espinosa-Guerra, attacks nonlinear servo problems by means of variablestructure control method.

The fifteen chapter, “Variable Structure Systems In Computational In-telligence” by Onder Efe, Okyay Kaynak, Xinghou Yu, gives a general intro-duction on how to make VSS and computational intelligence function in acomplementary manner to each other.

The last chapter entitled “Sliding Modes Control for Systems with FastActuators: Singularly Perturbed Approach” written by Leonid Fridman, re-visits sliding mode control based on singular perturbation approaches forsystems with both slow and fast dynamics.

We are sincerely grateful to the many contributors who have given theirvaluable time and expertise to this project, without whom, this collectionof contributions would not have been possible. Our thanks go to ProfessorM. Thoma for his support of this book proposal and Dr T. Ditzniger forhis assistance in the preparation of this book. We would like to particularlythank Mr Noel Patson, who helped a great deal in editing the book, takingcare of most painful editorial tasks.

VI Preface

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Special thanks to our families, Zhenwei, Iris Hong, Alice, Kettie, andElizabeth, for their support, devotion and patience.

Australia, Singapore, Xinghuo YuSeptember 2001 Jian-Xin Xu

VIIPreface

Contents

First Stage of VSS: People and Events . . . . . . . . . . . . . . . . . . . . . . . . 1Vadim I. Utkin

An Integrated Learning Variable Structure Control Method . . 33Jian-Xin Xu

Discrete-time Variable Structure Control . . . . . . . . . . . . . . . . . . . . . 57Katsuhisa Furuta, Yaodong Pan

Higher-Order Sliding Modes for the Output-Feedback Controlof Nonlinear Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Giorgio Bartolini, Arie Levant, Alessandro Pisano, Elio Usai

Variable Structure Systems with Terminal Sliding Modes . . . . . 109Xinghuo Yu, Man Zhihong

Adaptive Backstepping Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Ali Jafari Koshkouei, Russell E. Mills, Alan S.I. Zinober

Sliding Mode Compensation, Estimation and OptimizationMethods in Automotive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Ibrahim Haskara, Cem Hatipoglu, Umit Ozguner

On Quasi-optimal Variable Structure Control Approaches . . . . 175Jian-Xin Xu, Jin Zhang

Robust Control of Infinite-Dimensional Systemsvia Sliding Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Yuri Orlov

Sliding Modes Applications in Power Electronics andElectrical Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Asif Sabanovic, Karel Jezernik, Nadira Sabanovic

On the Development and Application of Sliding ModeObservers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253Christopher Edwards, Sarah K. Spurgeon, Chee Pin Tan

Multivariable Output-Feedback Sliding Mode Control . . . . . . . . 283Liu Hsu, Jose Paulo Vilela Soares da Cunha, Ramon R. Costa,Fernando Lizarralde

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Sliding Modes, Differential Flatness and IntegralReconstructors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Hebertt Sira-Ramırez, Victor M. Hernandez

On Robust VSS Nonlinear Servomechanism Problem . . . . . . . . . 343Vadim Utkin, B. Castillo-Toledo, A. Loukianov, O. Espinosa-Guerra

Variable Structure Systems Theory in ComputationalIntelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365Mehmet Onder Efe, Okyay Kaynak, Xinghuo Yu

Sliding Mode Control for Systems with Fast Actuators:Singularly Perturbed Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391Leonid M. Fridman

X Contents

First Stage of VSS: People and Events

Vadim I. Utkin

The Ohio State University, Columbus OH 43210, USA

Abstract. The objective of paper to present Variable Structure Control in histor-ical perspective from the late fifties along with related research. The author tries tosketch the events that gave birth and were associated with the further developmentof this interesting research area. In many books, survey papers the authors men-tion that the research in this area were initiated in the former Soviet Union about40 years ago and then the sliding mode control methodology has been receivingmuch more attention from the international control community within the last twodecades. In author’s opinion this paper is a good chance to describe what happenedbefore ”the last two decades”, to demonstrate the initial ideas and to mention theresearchers actively working at the initial stage. Their names and contributionsdeserve to be mentioned because the results were published mainly in Russian andpractically unknown to the colleagues outside the Soviet Union.

1 Introduction

The term “Variable Structure System” (VSS) first appeared in the late fifties.Naturally at the very beginning several specific control tasks for second-orderlinear and non-linear systems were tackled and advantages of the new ap-proach were demonstrated. Then the main directions of further research wereformulated. In the course of further development the first expectations of suchsystems were modified, their real potential has been revealed. Some researchtrends proved to be unpromising while the others, being enriched by newachievements of the control theory and technology, have become milestonesin VSS theory.

The paper is not a survey. The idea of preparing a survey seems to theauthor pointless, since at each stage of development of VSS theory, surveypapers were published with vast material had been accumulated in the areaby the time of publications.

From the point of the present day, much more interesting than survey in-formation is to make “travel in years” staring from original ideas and hopesto scientific arsenal of modern VSS theory and applications. It is of interestto establish bridges between initial and terminal stations of this travel. Ap-plication aspects of VSS is a topic of interest as well, since some of the mainVSS theory trends were initiated due to application problems. Mathematical,design and implementation aspects of the above topics constitute the subjectof this paper.

X. Yu and J.-X. Xu (Eds.): Variable Structure Systems: Towards the 21st Century, LNCIS 274, pp. 1−32, 2002. Springer-Verlag Berlin Heidelberg 2002

2 First Steps

Three paper were published by S. Emel’yanov in late fifties on feedback designfor the second-order linear systems ([1] was the first of them). The novelty ofthe approach was that the feedback gains could take several constant valuesdepending on the system state. Although the term “Variable Structure Sys-tem” was not introduced in the papers, each of the systems consisted of a setlinear structures and was supplied with a switching logic and actually wasVSS. The author observed that due to altering the structure in the courseof control process the properties could attained which were not inherent inany of the structures. For example the system consisting of two conservativesubsystems (Fig.1)

Fig. 1. Two conservative structures

Fig. 2. Asymptotically stable VSS

x = −kx (1)

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k =k1 if xx > 0k2 if xx < 0 k1 > k2 > 0.

becomes asymptotically stable due to varying its structure on coordinateaxes (Fig.2). Another way for stabilization is to find a trajectory in a stateplane of one of the structures with converging motion. Then the switchinglogic should be found such that the state reaches this trajectory for any initialconditions and move along it. If in the system

x = a2x− kx, a2 > 0,

there are two structures with k1 > 0 and k2 < 0 (Fig3,4), then such trajectoryexists in the second structure (straight line s = c∗x1 + x2 = 0, c∗ = a2/2 +√

a22/4− k2 in Fig.4). As it can be seen in Fig.5 variable structure system

with switching logic

Fig. 3. Unstable structure I

Fig. 4. Unstable structure II

3First Stage of VSS: People and Events

Fig. 5. Asymptotically stable VSS with monotonous processes

k =k1, if xs > 0,k2, if xs < 0. (2)

is asymptotically stable with monotonous processes. It is of interest thata similar approach was offered by A. Letov [2], but his approach impliedcalculation of the switching time as a function of initial conditions. As aresult any calculation error led to instability, therefore the system was called“conditionally stable”.

Starting from early sixties term “Variable Structure Control” appearedin titles of the papers by S. Emel’yanov and his colleagues. Interesting at-tempts were made to stabilize second-order nonlinear plants [3]. The plantsunder study were unstable with several equilibrium points and could not bestabilized by any linear control which was common for many processes ofchemical technology. The universal design recipe could hardly be developed,so for any specific case the authors tried to “cut and glue” different piecesof available structures such that the system turned to be globally asymp-totically stable. In Fig.6, Fig.7, Fig.8 the state planes (x1 = x, x2 = x) ofthree structures with linear feedback are shown. Equilibrium points of eachof them are unstable. Partitioning the state plane into six parts led to anasymptotically stable variable structure system (Fig.9). Note that some ofthe state trajectories are oriented towards switching line s = x2 + cx1 = 0 inthe above example (Fig.9). It means that that having reached this line thestate trajectory can not leave it and for further motion the state vector willbe on this line. This motion is called sliding mode. Sliding mode played thedominant role in the further development of VSS theory.Sliding mode mayappear in our second example on the switching line, if 0 < c < c∗ (Fig.10).Since the state trajectory coincides with the switching line in sliding modeits equation may be interpreted as the sliding mode equation

x+ cx = 0 (3)

The equation (3) is the ideal model of sliding mode. In reality due smallimperfections in a switching device (small delay, hysteresis, time constant)

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the control switches at a finite frequency and the state is not confined to theswitching line but oscillates in its small vicinity.

Fig. 6. Unstable nonlinear structure I

Fig. 7. Unstable nonlinear structure II

Three important facts should be underlined now:


Fig. 8. Unstable nonlinear structure III

Fig. 9. Asymptotically stable nonlinear VSS

Fig. 10. VSS with sliding mode

1. The original system is governed by a non-linear second order equation,the order of motion equation is reduced after sliding mode starts.

2. The motion equation is linear and homogenous.3. Sliding mode does not depend on the plant dynamics and determined by

parameter c selected by a designer.

The above examples served as a starting point and let us outline variousdesign principles offered at the initial stage of VSS theory development. Thefirst most obvious principle implies taking separate pieces of trajectories of theexisting structures and combining them together to get a good (in some sense)trajectory of the motion in a feedback system. The second principle consistsin seeking individual trajectories in one of the structures with the desireddynamic properties and designing the switching logic such that starting from

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some instant the state moves along one of these trajectories. And finally,design principle based on enforcing sliding modes in the surface where thesystem structure is varied or control undergoes discontinuities.

Unfortunately, the hopes associated with the first two approaches have notbeen justified; their applications have been limited to the study of severalspecific systems of low order. The promising control design principles arebased on enforcing sliding modes due to the properties observed in our second-order examples. As a result sliding modes have had, and are still having,an exceptional role in the development of VSS theory. Therefore the term“Sliding Mode Control” is often used in literature on VSS as more adequateto the nature of feedback design approach.

3 VSS in Canonical Space

3.1 Free Motion

The sliding mode control, demonstrated for the second-order system in Sec-tion 2, was generalized for linear SISO dynamic systems of an arbitrary orderunder the strong assumption that their behavior is represented in canonicalspace - space of output and its time derivatives:

xi = xi+1, i = 1, ..., n− 1

xn = −n∑

i=1

aixi + bu, ai, b are plant parameters, u is control input. (4)

Similarly to the second-order systems control was designed as piece-wiselinear function of system output

u = −kx1,

k =k1 if x1s > 0k2 if x1s < 0 (5)

with switching plane

s =n∑

i=1

cixi = 0, ci = const, cn = 1.

The deign method of VSS (4),(5) was developed after V. Taran joined theresearch team [4]. The methodology was preserved:

• Sliding mode should exist at any point of switching plane, then it is calledsliding plane.

• Sliding mode should be stable.• The state should reach the plane for any initial conditions.


Unfortunately the first and second requirements may be in conflict. On onehand, sliding mode exists if the state trajectories in the vicinity of the switch-ing plane are directed to the plane, or [5]

lims→+0

s < 0, lims→−0

s > 0. (6)

These conditions for our system are of form

ci−1 − aici

= cn−1 − an, i = 2, ..., n− 1. (7)

On the other hand coefficients ci in sliding mode equation

x(n−1)1 + cn−1x

(n−2)1 + ...+ c1x1 = 0 (8)

should satisfy Hurwitz conditions. The result of [4]: a sliding plane with stablemotion exists if and only if there exists k0, k2 < k0 < k1 such that the linearsystem with control u = k0x1 has (n−1) eigenvalues with negative real parts.

The result of [6]: for the state to reach a switching plane from any initialposition it is necessary and sufficient that the linear system with u = −k1x1

does not have real positive eigenvalues.The above results mean that for asymptotical stability of VSS system

each of the structures may be unstable. For example the third-order VSSsystem,

d3xdt3 = u, u = −kx, k =

1, if xs > 0,−1, if xs < 0. s = c1x+ c2x+ x,

consisting of two unstable linear structures is asymptotically stable for c1 = c22(sliding plane existence condition (7)), c1 > 0, c2 > 0 (Hurwitz condition forsliding mode). The reaching condition holds as well since the linear systemwith k = 1 does not have real positive eigenvalues.

Again sliding mode equation (8) is of a reduced order, linear, homoge-nous, does not depend on plant dynamics and determined by coefficients inswitching plane equation. This property looks promising when controllingplants with unknown time-varying parameters. Unfortunately control (5) isnot applicable for this purpose because the conditions for sliding plane toexist (7) need knowledge on the parameters ai.

For the modified version of VSS control

u = −n−1∑i=1

kixi, (9)

k =

k′i, if xis > 0,k′′i , if xis < 0.

The plane s = 0 is a sliding plane for any values of ci if

bk′i > maxai,an(ci−1 − ai − cn−1ci + anci)

bk′′i > minai,an(ci−1 − ai − cn−1ci + anci)

, a0 = 0, i = 1, ..., n− 1. (10)

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The design procedure of VSS (9) consisting of 2n−1 linear structures im-plies selection of switching plane or sliding mode equation (8) with desireddynamics and then coefficients k′i and ki” (10) such that s = 0 is a slidingplane (assuming that the ranges of parameter variations are known). Reach-ing this plane may be provided by increasing coefficient k′1. Sliding modewith the desired properties starts in the VSS after a finite time interval. Thetime, preceding the sliding mode, may be decreased by increasing the gainsin control (9).

Development of special methods is needed if the last equation in (4) de-pends on time derivatives of control, since trajectories in the canonical spacebecome discontinuous. Two approaches were offered by N.Kostyleva in theframework of the VSS theory: first, designing a switching surface with thepart of state variables, and, second, using a pre-filter in controller [7]. For theboth cases the conventional sliding mode with the desired properties can beenforced. Traces of the approaches may be found in modern publications.

3.2 Disturbance Rejection

The property of insensitivity of sliding modes to plant dynamics may beutilized to control plants subjected to unknown external disturbances. It isobvious that control (9) does not fit for this purpose. Indeed at the desiredstate (all xi are equal to zero) the control is equal to zero as well and unableto keep the plant there at presence of disturbances. We demonstrate how thedisturbance rejection problem can be solved using dynamic actuators withvariable structure. Let plant and actuator be integrators in the second-ordersystem (Fig.11). An external disturbance f(t) is not accessible for measure-ment.

The control is designed as a piece-wise linear function not only of theoutput x = x1 to be reduced to zero but also of actuator output y :

x1 = y + f(t)y = u,u = −kx1 − kyy.

orx1 = x2

x2 = −kx1 − kyx2 − kyf + f .

For the variable structure system with

k =

k0, if x1s > 0,−k0, if x1s < 0. , ky =

ky0, if ys > 0,−ky0, if ys < 0. , s = cx1 +x2, k0, kyo, c

are const.The state semi-planes x1 > 0 and x1 < 0 are shown in Fig.12. For domain

x2 < |f(t)| the singular points for each semi-plane are located in the oppositeone and as a result state trajectories are oriented toward switching line s = 0.It means that sliding mode occurs in this line with motion equation x+cx = 0and solution tending to zero. The effect of disturbance rejection may beexplained easily in structural language. Due to altering sign of local feedbackfor the actuator its output may be either diverging or converging exponential


function (Fig.13). In sliding mode due to high-frequency switching an averagevalue of the output is equal to the disturbance with an opposite sign. It is clearthat the disturbance, which can be rejected, should be between the divergingand converging exponential functions at any time. Similar approach standsbehind the disturbance rejection method in VSS of an arbitrary order.

Fig. 11. System with VS actuator

Fig. 12. Phase portrait of system with disturbance

3.3 Problem of Differentiation

Implementation of variable structure control algorithms studied in the previ-ous sections needs information on time derivatives of a system output. Since

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Fig. 13. Disturbance rejection in system with VS actuator

ideal differentiators can not be built, ideal sliding mode can not appear inthe systems with real differentiators. V. Taran [8] was the first who ana-lyzed the behavior of such VSS. The ideal derivative in the second-order VSSwas replaced by the output of the first-order filter with transfer functionG(p) = T1p+1

T2p+1 , (T1 and T2 are constant values):

x1 = x2

x2 = −a1x1 − a2x2 + bu, u = −kx1,

k =k1, if x1s > 0,k2, if x1s < 0. s = z1 −Qx1, z1(p) = G(p)x1(p) (11)

It was shown that the oscillations occurs in the system with a finite fre-quency (Fig. 14 and 15) instead of ideal sliding mode on the switching lines = 0. We believe that it was the first mentioning the phenomenon calledlately chattering which became the main obstacle for implementation of slid-ing modes in variable structure systems. The idea of chattering suppressionwas also offered in the paper by V.Taran [9]. Again the switching functionwas selected as a linear combination of the system output and output of afirst-order filter but the discontinuous control was an additional input of thefilter: the function s in (10) is replaced by s = cx1 − z, z = 1

T (z−αu), c, T, αare const.

Fig. 14. VSS with real differentiator


Fig. 15. Motion projection on plane (x1, z)

Since time-derivative s = cx2 − 1T (z − αu) is a discontinuous function of

s, the conditions (6) may be fulfilled and the sliding mode free of chatteringmay occur in some domain D (Fig.16(a)). This motion is governed by thesecond-order equations with the desired dynamics (Fig.16(b)) although theideal derivative is not used in the control algorithm.

Fig. 16. System with VS filter

Although generalization for arbitrary-order systems met serious analyt-ical difficulties, these publications first drew attention of the researchers inthe area to the chattering problem and outlined the approach to solve thisproblem.

3.4 Adaptive VSS

In all second-order examples, after sliding mode starts, the motion does notdepend on plant parameters and decays at the rate determined by the coef-ficient c in the equation (3) (of course c should be positive). For the system

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(4) with n = 2, ai = 0 and control (5) the conditions (6) for sliding mode toexist on the switching line s = cx1 + x2 = 0 are of form bk2 < −c2 < bk1. Ifparameter b is unknown and varies in known range 0 < bmin < b < bmax andthe feedback gain is bounded |k| < k0 then the line s∗ = c0x1 + x2 is alwaysa sliding line for c0 =

√k0bmin and k1 = −k2 = k0.Then after reaching the

line at some time t1, the system output will decay as exponential functionx1(t1)e−c0(t−t1) for any value of b.

Fig. 17. Adaptive VSS with state dependent switching line

However the rate of decay may be increased if b takes any values differentfrom bmin by increasing gain c. It should be done without measuring b. Wedescribe the adaptation approach offered by E.Dubrovsky [10]. While slidingmode exists the gain c is increased until sliding mode disappears. The factof sliding mode existence may easily established by averaging discontinuouscoefficient k taking two values k0 or −k0 and switching at high frequency. If|kav| < k0 then sliding mode exists, if |kav| is approaching k0 then slidingmode is about to disappear and further increasing c should be terminated.As a result a sliding line with maximal value of c or maximal rate of solutiondecay for a current unknown value of coefficient b is found (Fig.17).

3.5 Preliminary Mathematical Remark

The basic design idea of the first stage of VSS theory was to enforce slidingmode in a plane of the canonical space. An equation of a sliding plane de-pending a system output and its derivatives was interpreted as a sling modeequation. Formally this interpretation is not legitimate. “ To solve differentialequation” means to find a time function such that substitution of the solutioninto the equation makes its right- and left-hand sides equal identically.

In our second-order example with control (2) and 0 < c < c∗, slidingmode existed on the switching line s = 0 and equation (3) was taken as thesliding mode equation. Its solution Ae−ct being substituted into function s


makes it equal to zero. Control (2) is not defined for s = 0 and respectivelythe right-hand side of the system equation (1) is not defined as well. There-fore we cannot answer the question whether the solution to (2) is the solutionto the original system. One of the founders of control theory in the USSRacademician A.A. Andronov indicated that ambiguity in the system behavioris eliminated if minor non-idealities such as time delay, hysteresis, small timeconstants are recognized in the system model which results in so-called realsliding mode in a small neighborhood of the discontinuity surface. Ideal slid-ing motion is regarded as a result of limiting procedure with all non-idealitiestending to zero [11]. The examples of such limiting procedure were also given.The relay second-order system was considered with motion equations in thecanonical state and a straight line (3) as a set of discontinuity points forcontrol. The behavior of the system was studied under the assumption that atime delay was inherent in the switching device and, consequently, the discon-tinuity points were isolated in time. It was found that with the delay tendingto zero irrespective of the plant parameters and disturbances, the solution ofthe second-order equation in sliding mode always tended to the solution ofthe first order equation (3) which depends only on the gain c of the switchingline equation. The validity of the equation (8) as the model of sliding modein the canonical space of an arbitrary-order system may be substantiated inthe similar way.

3.6 Comments for VSS in Canonical Space

In the sixties VSS studies were mainly focused on linear (time-invariant andtime-varying) systems of an arbitrary order with scalar control and scalarvariable to be controlled. These first studies utilized the space of an errorcoordinate and its time derivatives, or canonical space, while the controlwas designed as a sum of state components and accessible for measurementdisturbances with piece-wise constant gains. When the plant was subjected tounknown external disturbances, local feedback of the actuators was designedin the similar way. As a rule, the discontinuity surface was a plane in thecanonical space or in an extended space, including the states of the filtersfor deriving approximate values of derivatives. In short, most works of thisperiod on VSS treated piece-wise linear systems in the canonical space withscalar control scalar controlled coordinate. The invariance property of slidingmodes in the canonical space to plant dynamics was the key idea of all designmethods.

The first attempts to apply the results of VSS theory demonstrated thatthe invariance property of sliding modes in canonical spaces had not beenbeneficial only. In a sense it decelerated development of the theory. The il-lusion that any control problems may be easily solved should sliding modebe enforced led to some exaggeration of sliding mode potential. The fact isthat the space of derivatives is a mathematical abstraction, and in practiceany real differentiators have denominators in transfer functions; so the study

14 V.I. Utkin

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of the system behavior in the canonical space can prove to be unacceptableidealization. Unfortunately the attempts to use filters with variable structurefor multiple differentiation have not led to any significant success.

For the just discussed reason, research of the fairly narrow class of VSSmainly carried out at Institute of Control Sciences, Moscow and by a groupof mathematicians headed by E. Barbashin at Institute of Mathematics andMechanics, Sverdlovsk, did not result in wide applications and did not pro-duced significant echo in the scientific press. The results of this first stage ofVSS development, i.e. analysis and design of VSS in the canonical space, weresummarized in [12] and [5]. In view of the limited field of practical applica-tions in the frame of this approach (VSS with differentiating circuits), anotherextreme appeared, reflecting certain pessimism about implementation of anyVSS with sliding modes.

The second stage of development of VSS theory began roughly in the latesixties and early seventies, when design procedures in the canonical spaceceased to be looked as obligatory and studies were focused on systems ofgeneral form with arbitrary state components and with vector control andoutput values. The first attempts to revise VSS methodology in this newenvironment demonstrated that the pessimism about sliding mode controlwas unjustified and refusal to utilize the potential of sliding mode was anunreasonable extravagance.

4 Problem Statements

At the second stage, from early seventies to the present, the range of theproblems discussed in the context of VSS theory was essentially widened. Atransition was made from canonical to more general state space, systems withvector control were under study, the dynamic plant to be controlled becameessentially nonlinear, the surfaces where the system structure was alteredbecame nonlinear as well. Development of VSS theory for the new class ofproblems demanded creation of qualitatively new mathematical and designmethods.

We assume that for the class of the systems under study each componentof control vector can be equal to one of two continuous functions of state andtime and the structure of the system is altered at surfaces in the system statespace:

x = f(x, t, u), (12)

where x ∈ Rn is a state vector, u ∈ Rm is a control vector. The control isdesigned as a discontinuous function of the state such that each componentundergoes discontinuities in some surface in the system state space:

ui =u+i (x, t), if si(x) > 0,

u−i (x, t), if si(x) < 0.

, i = 1, ...,m (13)


where f(x, t, u), u+i ( x ,t), u−

i (x, t) and si(x) are continuous functions oftheir arguments u+

i ( x ,t) = u−i (x, t).

Similar to the above examples of VSS in the canonical space, state velocityvectors may be directed towards one of the surfaces and sliding mode arisesalong it (arcs ab and cb in Fig.18). It may arise also along the intersection oftwo of the surfaces (arc bd). Fig.19 illustrates the sliding mode in the inter-section even if it does not exist at each of them taken separately. For generalcase (12) and (13) sliding mode may exist in the intersection of all disconti-nuity surfaces si = 0, or in the manifold s(x) = 0, sT (x) = [s1(x), ..., sm(x)],

s ∈ n−m.

Fig. 18. Two-dimensional sliding mode

Fig. 19. Sliding mode in intersection of switching surfaces only

Let us discuss what kind of benefit sliding modes may result in, should thismotion be enforced in the control system. First, in sliding mode the output sof the element implementing discontinuous control is close to zero, while its

16 V.I. Utkin

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output (exactly speaking its average value uav) takes finite values (Fig.20).Hence the element implements high (theoretically infinite) gain, that is theconventional tool to reject disturbance and other uncertainties in the systembehavior. Unlike to systems with continuous controls, this property calledinvariance is attained using finite control actions. Second, since sliding modetrajectories belong to a manifold of dimension lower than that of the originalsystem, the order of the system is reduced as well. This enables a designerto simplify and decouple the design procedure. Both order reduction andinvariance are transparent for the above two second-order systems.

Fig. 20. High gain implementation in sliding mode

In order to justify the above arguments in favor of using sliding modes incontrol systems, we, first, need mathematical methods for deriving equationsof sliding modes in the intersection of discontinuity surfaces and, second,the conditions for the sliding mode to exist should be obtained. Only havingthese mathematical tools the design methods of sliding mode control for widerange of control problems may be developed.

Sliding mode equations, the existence condition and design are base stonesof sliding mode control theory. Of course this research scope should be con-sidered along with application issues.

5 Sliding Mode Equations

The first mathematical problem concerns differential equations of slidingmode. For our second-order examples this equation was obtained using heuris-tic approach: the equation of the switching line x+ cx = 0 was interpreted asthe motion equation. But even for an arbitrary time invariant second-orderrelay system

x1 = a11x1 + a12x2 + b1ux2 = a21x1 + a22x2 + b2u,u = −Msign(s), s = cx1 + x2; M,aij , bi, c are const.

(14)


the problem does not look trivial since in sliding mode s = 0 is not a motionequation.

The problem of mathematical description of sliding mode is quite a chal-lenge and requires the design of special techniques. It arises due to discontinu-ities in control, since the relevant motion equations with discontinuous right-hand sides do not satisfy the conventional theorems on existence-uniquenessof solutions (Lipshitz constant does not exists for discontinuous functions).

Uncertainty in behavior of discontinuous systems on the switching sur-faces gives freedom on choosing an adequate mathematical model and gavebirth to a number of lively discussions. For example, description of slidingmode in system (14) by a linear first-order differential equation, a commonpractice today, seemed unusual at first glance; this model was offered by Yu.Kornilov [13] and A. Popovski [14] and the approach was generalized forlinear systems of an arbitrary order by Yu. Dolgolenko [15] in fifties.

The approach by A.Filippov [16], now recognized as classical, was notaccepted by all experts in the area: at the 1st IFAC congress in 1960 Yu.Neimark (author of one more model of sliding mode based on convolutionequation) offered an example of a system with two relay elements with asolution different from that of Filippov’s method [17]. The discussion at thecongress proved to be fruitful from the point of stating a new problem in thetheory of sliding modes. The discussion led to the conclusion that two dy-namic systems with identical equations outside a discontinuity surface mayhave different sliding mode equations. Most probably the problem of un-ambiguous description of sliding modes in discontinuous systems was firstbrought to light.

In situations when conventional methods are not applicable, the usual ap-proach is to employ regularization or replacing the initial problem by a closelysimilar one, for which familiar methods can be used. In partice, taking intoaccount delay or hysteresis of a switching element, small time constants inan ideal model, replacing a discontinuous function by a continuous approxi-mation are examples of the regularization since discontinuity points (if theyexist) are isolated. As we discussed in Section 3.5, such regularization withtime delay was employed by Andronov for substantiation of sliding modeequation (3). Similar approach for nonlinear relay system with imperfectionsof time delay and hystetresis type was developed by J.Andre and P.Seibert[18]; it is interesting that their sliding mode equations coincided with thoseof Filippov’s method and the result may serve as substantiation of Filippov’smethod. At the same time the question may be asked, whether the methodis applicable for the systems with different types of non-idealities. Generallyspeaking the answer is negative: a continuous approximation of a discontin-uous function leads to motion equations different from those of Filippov’smethod [19]. So we should admit that for systems with a right-hand side as anonlinear function of discontinuous control (12),(13) sliding mode equationscan not be derived unambiguously even for the case of scalar control. The

18 V.I. Utkin

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“point-to-point” technique used in the cited here papers for scalar case isnot applicable for the system with vector control and sliding modes in theintersection of a set of discontinuity surfaces.

The universal approach to regularization consists of introducing a bound-ary layer ‖s‖ < ∆, ∆ − const around manifold s = 0, where an ideal dis-continuous control is replaced by a real one such that the state trajectoriesare not confined to this manifold but run arbitrarily inside the layer. Thenonidealities, resulting in motion in the boundary layer, are not specified,the only assumption for this motion is that the solution exists in the con-ventional sense. If, with the with of the boundary layer ∆ tending to zero,the limit of the solution exists, it is taken as a solution to the system withideal sliding mode. Otherwise we have to recognize that the equations be-yond discontinuity surfaces do not derive unambiguously equations in theirintersection, or equations of the sliding mode.

The boundary layer regularization enables substantiation of so-called Equiv-alent Control Method intended for deriving sliding mode equations in mani-fold s = 0 if the system (12) is linear with respect to control

x = f(x, t) +B(x, t)u (15)

with B(x, t) being n×m full rank matrix

G(x)B(x), G(x) = ∂s/∂x, det(GB) = 0.

Following this method the sliding mode equation with a unique solution mayderived for nonsingular matrix.

First, the equivalent control should be found as the solution to the equa-tion s = 0 on the system trajectories (G and (GB)−1 are assumed to exist):

s = Gf +GBu = 0, ueq = -( GB ) −1Gf. (16)

Then the solution should be substituted into (15) for the control

x = f −B(GB)−1Gf (17)

Equation (17) is the sliding mode equation with initial conditions

s(x(0), 0) = 0.

Since s(x) = 0 in sliding mode m components of the state vector maybe found as a function of the rest (n − m) ones: x2 = s0(x1); x2, s0 ∈ m;x1 ∈ n−m and, correspondingly, the order of sliding mode equation may bereduced by m:

x1 = f1[x1, t, s0(x1)], f1 ∈ n−m (18)

The idea of the equivalent control method may be easily explained withthe help of geometric consideration. Sliding mode trajectories lie in the man-ifold s = 0 and the equivalent control ueq being a solution to the equation


s = 0 implies replacing discontinuous control by such continuous one thatthe state velocity vector lies in the tangential manifold and as a result thestate trajectories are in this manifold. It will be important for control designthat sliding mode equation (18)

• is of reduced order• does not depend on control• depends on the equation of switching surfaces.

6 Sliding Mode Existence Conditions

The second mathematical problem in the analysis of sliding mode as a phe-nomenon is deriving the conditions for sliding mode to exist. As to the systemswith scalar control the conditions may be obtained from geometrical consid-erations: the deviation from the switching surface s and its time derivativeshould have opposite signs in the vicinity of a discontinuity surface s = 0 (6).

As it was demonstrated in the example in Fig.14, for existence of slid-ing mode in an intersection of a set of discontinuity surfaces si(x) = 0,(i = 1, ...,m) it is not necessary to fulfill inequalities (6) for each of them.The trajectories should converge to the manifold sT = (s1, ..., sm) = 0 andreach it after a finite time interval similarly to the systems with scalar control.The term “ converge” means that we deal with the problem of stability of theorigin in m-dimensional subspace (s1, ..., sm), therefore the existence condi-tions may be formulated in terms of the stability theory. The non-traditionalcondition: finite time convergence should take place. This last condition isimportant to distinguish the systems with sliding modes and the continuoussystem with state trajectories converging to some manifold asymptotically.For example the state trajectories of the system x − x = 0 converge to themanifold s = x−x = 0 asymptotically since s = −s, however it would hardlybe reasonable to call “sliding mode” the motion in s = 0.

Further we will deal with the conditions for sliding mode to exist for affinesystems(15). To derive the existence conditions, the stability of the motionprojection on subspace s

s = Gf +GBu (19)

should be analyzed. The control (13) may be represented as

u(x, t) = u0(x, t) + U(x, t)sign(s)

with

u0(x) =u+(x,t)+u−(x,t)

2 ,

diagonal matrix U with elements

Ui =u+

i(x,t)−u−

i(x,t)

2 , i = 1, ...,m

20 V.I. Utkin

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and

[sign(s)]T = [sign(s1), ..., sign(sm)]

Then the motion projection on subspace s is governed by

s = d(x)−D(x)sign(s), (20)

with d = Gf +GBu0, D = −GBU .To find the stability conditions of the origin s = 0 for nonlinear system

(20), or the conditions for sliding mode to exist, we will follow the stan-dard methodology for stability analysis of nonlinear systems – try to find aLyapunov function.

Definition 1. The set S(x) in the manifold s(x) = 0 is the domain of slidingmode if for the motion governed by equation (20) the origin in the subspaces is asymptotically stable with finite convergence time for each x from S(x).

Definition 2. Manifold s(x) = 0 is referred to as sliding manifold if slidingmode exists at its each point, or S(x)=x : s(x) = 0..

Theorem 1. S(x) is a sliding manifold for the system with motion projectionon subspace s governed by s = −Dsign(s), if matrix D + DT is positivedefinite.

Theorem 2. S(x) is a sliding manifold for system (20) if D(x) +DT (x) > 0,λ0 > d0

√m, λmin(x) > λ0 > 0, ‖d(x)‖ < d0, λmin is the minimal eigenvalue

of matrix D+D2

T, λmin > 0.

The statements of the both the theorems may be proven using sum ofabsolute values of si

V = sT sign(s) > 0

as a Lyapunov function.

7 Design Principles

7.1 Decoupling and Invariance

The above mathematical results constitute the background of the designmethods for sliding mode control involving two independent subproblemsof lower dimensions:

• design of the desired dynamics for a system of the (n−m)th order by aproper choice of a sliding manifold s = 0 ;


• enforcing sliding motion in this manifold which is equivalent to stabilityproblem of the mth order system.

All the design methods will be discussed for affine systems (15) which arelinear with respect to control but not necessary with respect to state. Sincethe principle operating mode is in the vicinity of the discontinuity points, theeffects inherent in the systems with infinite feedback gains may be obtainedwith finite control actions. As a result sliding mode control is an efficient toolto control dynamic high-order nonlinear plants operating under uncertaintyconditions (e.g. unknown parameter variations and disturbances).

Formally the sliding mode is insensitive, or invariant to “uncertainties”in the systems satisfying the matching conditions h(x, t) ∈ range(B) by B.Drazenovic, with vector h(x, t) characterizing all factors in a motion equation

x = f(x, t) +B(x, t)u+ h(x, t)

whose influence on the control process should be rejected. Matching condi-tion means that disturbance vector h(x, t) may be represented as a linearcombination of columns of matrix B [20].

7.2 Regular Form

The design procedure may be illustrated easily for the systems representedin the Regular Form

x1 = f1(x1, x2, t), x1 ∈ Rn−m

x2 = f2(x1, x2, t) +B2(x1, x2, t)u, x2 ∈ Rm, det(B2) = 0. (21)

The state subvector x2 is handled as a fictitious control in the first equa-tion of (21) and selected as a function of x1 to provide the desired dynamicsin the first subsystem (the design problem in the system of the (n − m)th

order with m-dimensional control):

x2 = −s0(x1).

Then the discontinuous control should be designed to enforce sliding modein the manifold

s(x1, x2) = x2 + s0(x1) = 0 (22)

(the design problem of the mth order with m-dimensional control).After a finite time interval sliding mode in the manifold (22) starts and

the system will exhibit the desired behavior governed by

x1 = f1[x1,−s0(x1), t].

22 V.I. Utkin

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Note that the motion is of a reduced order and depends neither functionf2(x1, x2, t) nor function B2(x1, x2, t) in the second equation of the originalsystem (21).

Since the design procedures for the systems in Regular Form (21) aresimpler than for those (16) it of interest to find the class of systems (16)for which a nonlinear coordinate transformation exists such that it reducesthe original system (15) to the form (21). By the assumption rank(B) = mtherefore (15) may be represented as

x1 = f1(x1, x2, t) +B1(x1, x2, t)u, x1 ∈ Rn−m

x2 = f2(x1, x2, t) +B2(x1, x2, t)u, x2 ∈ Rm, det(B2) = 0.

For coordinate transformation

y1 = ϕ(x, t), y2 = x2 , y1, ϕ ∈ n−m,

the equation for y1

y1 = ∂ϕ∂x f + ∂ϕ

∂xBu+ ∂ϕ∂t

does not depend on control u, if vector function ϕ is a solution to matrixpartial differential equation

∂ϕ

∂xB = 0. (23)

Then the motion equations with respect to y1 and x2 are in the regularform. Necessary and sufficient conditions of solution existence and uniquenessfor (23) may be found in terms of Pfaff ‘s forms theory and Frobenius theoremwhich constitute a well developed branch of mathematical analysis [21].

7.3 Enforcing Sliding Modes

At the second stage of feedback design, discontinuous control should be se-lected such that sliding mode is enforced in manifold s = 0. As shown inSection 6, sliding mode will start at manifold s = 0 if the matrix GB+(GB)T

is positive-definite and the control is of the form

u = − M(x, t) sign(s) (component-wise)

with a positive scalar function M(x, t) chosen to satisfy the inequality

M(x, t) > λ−1 ‖Gf‖,where λ is the lower bound of the eigenvalues of the matrix GB + (GB)T .

Now we demonstrate a method of enforcing sliding mode in manifold s = 0for an arbitrary nonsingular matrix GB.The motion projection equation onsubspace s may be written in the form

s = GB(u− ueq).


Remind that the equivalent control ueq is the value of control such that timederivative of vector s is equal to zero.

Let V = 12s

T s be a Lyapunov function candidate and the control be ofform

u = −M(x, t)sign(s∗), s∗ = (GB)T s.

Then V = −s∗T sign(s∗)+s∗T ueq

M(x,t) and V is negative definite for M(x, t) >‖ueq‖.

It means that sliding mode is enforced in manifold s∗ = 0 which means itsexistence in manifold s = 0 selected on the first step of the design procedure.It is important that the conditions for sliding mode to exist are inequalitiestherefore upper estimate of the disturbances is needed rather than preciseinformation on their values.

7.4 Unit Control

The Lyapunov method approach implies design of control based on Lyapunovfunction selected for a nominal (feedback or open loop) system. The objectiveis to find the control such that a time-derivative of the Lyapunov functionis negative on the trajectories of the system with perturbations caused byuncertainties of a plant operator and environment conditions. The roots ofthe above approach may be found in papers by G. Leitmann and S. Gutmanpublished in 70’s [22] and it leads to discontinuous control and sliding modes.

Now we demonstrate how this approach can be applied to enforce slid-ing mode in manifold s = 0, selected in compliance with some performancecriterion at the first stage of feedback design for the system with unknowndisturbance vector h(x, t).The control is designed as a discontinuous functionof s

u = −ρ (x, t)DT s (x)‖DT s (x)‖ (24)

with D = ∂s/∂xB, D is assumed to be nonsingular. Note that the normof control (24) with the gain is equal to 1 for any value of the state vector.It explains the term “unit control“ for control (24).

The equation of a motion projection of the system with disturbance h(x, t)on subspace s is of form

s =

∂ϕ∂x

(f + h) +Du.

The conditions for the trajectories to converge to the manifold s (x) = 0and sliding mode to exist in this manifold may be derived based on Lyapunovfunction

V =12sT s (25)

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with time derivative

V = sT∂s∂x

(f + h)− ρ (x, t)

∥∥DT s (x)∥∥

<∥∥DT s (x)

∥∥ · [∥∥D−1∂s∂x

(f + h)

∥∥ − ρ (x, t)].

(26)

For ρ (x, t) >∥∥D−1

∂s∂x

(f + h)

∥∥ the value of V is negative thereforethe state will reach the manifold s (x) = 0 after a finite time interval forany initial conditions and then the sliding mode with the desired dynamicsoccurs. Finiteness of the interval preceding the sliding motion follows frominequality resulting from (25), (26)

V < −γV12 , γ = const > 0

with the solution

V (t) <(−γ

2 t+√Vo

)2, Vo = V (0) .

Since the solution vanishes after some ts < 2γ

√Vo, the vector s vanishes

as well and the sliding mode starts after a finite time interval.It is of interest to note the principle difference in motions preceding the

sliding mode in s (x) = 0 for the conventional component-wise control andunit control design methods. For the conventional method the control un-dergoes discontinuities should any of the components of vector s changessign, while the unit control is a continuous state function until the manifolds (x) = 0 is reached. By this reason unit control systems with sliding modeswould hardly be recognized as VSS.

8 Discrete-Time Sliding Mode Control

Sliding mode control was first developed for continuous time systems gov-erned by finite-dimensional differential equations. Most of modern controlsystems are usually based on discrete micro-controller implementation. How-ever, discontinuous control designed for a continuous-time system leads tohigh frequency oscillations referred to as chattering since within a samplinginterval control is constant and switching frequency cannot exceed that ofsampling. Increasing a sampling frequency to decrease the chattering am-plitude seems unjustified. We believe that using a computer is adequate tocontrol system dynamics if a sampling frequency corresponds to average, slowsystem motion rather than to a high frequency component.

The state trajectories in discrete-time systems with discontinuous controlare not confined to the switching manifold but to some domain around it.So the first approach is to design control such that qualitatively this motionhas the same properties as its continuous-time counterpart. To reduce thisdomain the control may be designed as a sum of continuous state functionand a discontinuous one with lower magnitude.

The term “sliding mode” was first introduced for continuous time equa-tions. For discrete-time equations


xk+1 = F (xk, uk), uk = u(xk), xi ∈ n, u ∈ m

this term should be newly introduced in the spirit of Definitions of Section6.

Definition 3. The set S(x) in the manifold s(x) = 0, s ∈ m is thedomain of sliding mode if there exists vicinity ε of S such that s(xk+1) =s[F (xk, uk)] = 0 for xk ∈ ε.

The solution to the equation in Definition 3 is called equivalent controlukeq as well, since, similar to continuous systems, it results in motions withstate trajectories in the manifold s(x) = 0.

The discrete-time sliding mode control with bounded control actions ‖u‖ ≤u0 is of form

uk =

ukeq, if ‖ukeq‖ ≤ u0ukeq

‖ukeq‖u0, if ‖ukeq‖ > u0

Both discrete-time control and the equivalent control are continuous statefunctions. For example the equivalent control is linear state function ukeq =−(CB)−1CAxk for linear time-invariant discrete-time systems xk+1 = Axk+Buk with a linear sliding manifold sk = Cxk = 0.

The control ukeq can not be found for linear plants with unknown param-eters in matrix A, then the modified version should be applied

uk = −(CB)−1sk, if

∥∥(CB)−1sk∥∥ ≤ u0

−u0(CB)−1sk/∥∥(CB)−1sk

∥∥ , if∥∥(CB)−1sk

∥∥ > u0.

For the both versions the control system is free of chattering and themotion equation is of a reduced order. The accuracy of the systems operatingunder uncertainty conditions is of a sampling interval order.

Similar to continuous-time systems, the motion with state trajectories ina manifold

s(x) = 0, s ∈ Rm

and finite time needed to reach the manifold may occur in discrete-timesystem as well. The fundamental difference is that the control should be acontinuous function of the state.

9 Chattering Problem

As we discussed in Section 3.3, replacement of the ideal time derivative inthe switching line equation by a first-order filter led to oscillations at a finitefrequency in some vicinity of the line. These oscillations called chattering mayoccur in the systems of general type (13), (15) at the presence of unmodeleddynamics such as disregarded in the ideal model small time constants ofactuators and sensors. (µ1 and µ2 in Fig.16 with a linear plant and z1 andz2 as unmodeled-dynamics state vectors).

26 V.I. Utkin

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Chattering is a harmful phenomenon because it leads to low system accu-racy, high wear of mechanical moving parts, high heat losses in power electriccircuitry. Chattering is considered as the main reason which hinders appli-cation of sliding mode control since dynamic mismatch between real plantsand its model always exists.

Continuous approximation of discontinuous control in a boundary layerof a sliding manifold is a method of chattering suppression if the unmodeleddynamics is not excited. At the same time chattering may appear in systemwith unmodeled dynamics and continuous approximation of control if thegain in a boundary layer is too high. Since the values of time constants,neglected in an ideal model, are unknown, the design should be oriented tothe worst case and as a result the invariance property of discontinuous controlare not utilized to full extent.

Further studies and practical experience showed that the chattering causedby unmodeled dynamics may be eliminated in systems with asymptotic ob-servers (Fig. 21).

Fig. 21. Chattering suppression in systems with observers

In spite of the presence of unmodeled dynamics, ideal sliding arises, it isdescribed by a singularly perturbed differential equation with solutions freeof a high-frequency component and close to those of the ideal system. Asshown in Figure 21. the asymptotic observer serves as a bypass for the high-frequency component, therefore the unmodeled dynamics are not excited.Note that a trace of the idea “a bypass for the high-frequency component”may be found in the cited in Section 3.3 publications of the sixties.

To design the observers system parameters are needed, but sliding modeis not destroyed if a priori knowledge of parameters differs from their real


values. The sensitivity of state estimation to parameter variations may bereduced in observers with high gains. Preserving sliding modes in systemswith asymptotic observers predetermined successful applications of slidingmode control.

10 Application

The experience that has been gained in the applications of sliding modecontrol validates their efficiency for a wide range of processes in moderntechnology. As we mentioned in Introduction this paper does not claim tobe a survey on VSS theory. The objectives of the paper neither implies tosurvey the results of applications which have always been associated withtheoretical studies. We confine ourselves to one example only – results of theinternational project of colleagues from Yugoslavia and the Soviet Union inseventieth on sliding mode control of induction motors. The reason is twofold:

• It was the first application of multi-dimensional sliding mode to high-order nonlinear plants with control algorithms given in Section 7 of thispaper

• Control of electric drives is the most challenging application of slidingmodes; implementation of sliding modes by means of the most commonelectric power components has turned out to be simple enough, since “on-off” is the only admissible mode for them and discontinuities in controlare dictated by the very nature of power converters.

The behavior of an induction motor is described by a nonlinear high-ordersystem of differential equations:

dndt = 1

J (M −ML) , M = xH

xR(iαφβ − iβφα)

dφα

dt = − rR

xRφα − nφβ + rR

xH

xRiα,

dφβ

dt = − rR

xRφβ + nφα + rR

xH

xRiβ

diαdt = xR

xsxR−x2H

(−xH

xR

dφα

dt − rsiα + uα

)diβdt = xR

xsxR−x2H

(−xH

xR

dφβ

dt − rsiβ + uβ

)u = (uα, uβ)

T = 23 (eR, eS , eT ) (uR, uS , uT )

T

(27)

where n is a rotor angle velocity, and two-dimensional vectors φT = (φα, φβ);iT = (iα, iβ), uT = (uα, uβ) are rotor flux, stator current and voltage inthe fixed coordinate system (α, β), respectively; M and ML are a torquedeveloped by a motor and a load torque, uR, uS , uT are phase voltages,which may be made equal either to u0 or −u0; eR, eS , eT are unit vectors ofphase windings, R, S, T ; J , xH , xS , xR, rR, rS are motor parameters.

The control goal is to make one of the mechanical coordinates, for ex-ample, an angle speed n (t), be equal to a reference input n0 (t) and themagnitude of the rotor flux ‖φ (t)‖ be equal to its scalar reference inputφ0 (t). The deviations from the desired motions are described by the function

28 V.I. Utkin

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s1 = c1 [n0 − n (t)] + ddt [n0 − n (t)], s2 = c2 [φ0 − ‖φ (t)‖] + d

dt [φ0 − ‖φ (t)‖],where c1, c2 are positive. The static inverter forms three independent controlsuR, uS , uT so one degree of freedom can be used to satisfy some additional cri-terion. Let the voltages uR, uS , uT constitute a three-phase balanced system,which means that the equality s3 =

∫ t

0(uR + uS + uT ) dt = 0 should hold for

any t. If all three functions s1, s2, s3 are equal to zero then, in addition tobalanced system condition, the speed and flux mismatches decay exponen-tially since s1 = 0, s2 = 0 with c1, c2 > 0 are first-order differential equations.This means that the design of three dimensional control uT = (uR, uS , uT ) isreduced to enforcing the sliding mode in the manifold s = 0, sT = (s1, s2, s3)in the system (27). Control algorithm of Section 7.3 with a new set of dis-continuity surfaces s∗ = 0 was implemented for an experimental setup with11kw induction motor.

In the further research sliding mode observers were developed to finda rotor speed and flux simultaneously by measuring stator currents only.Later on different types of electric motors with sliding modes were applied inrobotics, metal-cutting machine tools, and electric vehicles.

11 Conclusions

The main attention in this paper was paid to the events and ideas of VSStheory before the last two decades, when the interest to VSS has been in-creased significantly and the further development has been associated withinternational cooperation of colleagues from research centers of many coun-tries. It can be easily revealed from intensity of publications in the area andthe number of special sessions and issues of the most prestigious conferencesand journals on control. “Current state of the art assessment” is beyond thescope of this paper. At the same time brief information on what recently ini-tiated research directions look promising would be helpful for understandingthe role of the “old times” in VSS history.

Simplex Control The system with control (13) consists of 2n continuousstructures. However if we give up component-wise design and select the setof control vectors directly, then it is sufficient to have n+ 1 vectors only forenforcing sliding mode in the intersection of surfaces. Such method is calledsimplex control design. It eliminates redundancy and ambiguity of chatteringor motion in a boundary layer that is important for multi-phase electricmachines.

Observers In contrast to the conventional state observers, the input of slid-ing mode observers is a discontinuous function of an error between a systemoutput and its estimate. As a result the order of error equations is reducedand the estimate does not depend on disturbances under the matching condi-tions, or if the disturbances are in the observer input space. For discrete-time


observers, deviation from a sliding manifold may used for estimation bothsystem state and parameters.

Adaptive Control In addition to the adaptation method described in Sec-tion 3.4, the model reference approach associated with sliding modes may beused for designing an adaptive system. Again the adaptation process doesnot depend on disturbances under the matching conditions.

Output Sliding Mode Control The state vector is not available in manyapplications. An alternative to systems with observers is designing controlas a function of the output only. As to linear systems, the requirements foreach structure are weaker than those for a feedback system to be designedand the order of the motion equations is reduced. The similar situation tookplace for sliding mode control in linear systems studied in the first sectionsof this paper.

Systems with Delays The concept “sliding mode” for the systems withdelays should be revised. It can be done in the spirit of the definition givenfor discrete-time systems in Section 8. Similar to what was done for discrete-time systems, the control may be designed such that system trajectories areconfined to a manifold of a dimension lower than that of the original system.

Infinite-Dimensional Systems The mathematical models of many moderntechnological processes are partial differential equations, which are particularcase of infinite-dimensional systems. The attempts of generalization of VSSmethodology to this class showed that even the basic concepts such as slidingmode, discontinuity surface, component-wise design were to be revised. Hemain obstacle which hinders utilizing sliding mode control is unboundnessof the operators in motion equations and the idea of enforcing sliding modeswith bounded control is not applicable. The recently developed mathematicaland design methods demonstrated that the potential of sliding mode controlmay utilized to full extent in infinite-dimensional systems as well.

The above list may be complemented by new design methods: second-order sliding mode control; VSS control of the systems in the form

f(x, x, u, u, ..., u(k), t) = 0;

sliding modes with finite transient times; integral sliding mode control withno reaching phase preceding sliding mode; robotics oriented methods incor-porating dynamics of drives and manipulators in design procedures.

Finally, a little fantasy : The author of Minimum Principle Pontryaginmentioned in 1953 that the systems with “glued trajectories” formally cannot be called dynamic systems [23]. The evolution of state x(t) in dynamicsystems is represented by shift operator x(t) = F [x(0), t] with unique val-ues for any time t (t may belong to or Z to embrace continuous- anddiscrete-time systems). Then the one parameter set of operators F [•, t] is

30 V.I. Utkin

g p

called “group” . However it is not the case for systems with sliding modesfor negative time t. Indeed a point in a sliding manifold may be reached insliding mode or from outside. It means that operator F does not have inverseand the set F [•, t] is called semigroup.

Then a new definition can be offered: the point x of the system x(t) =F [x, t] is called “sliding mode point” if the set F [x, t] constitutes a semi-group. All the definitions offered previously for continuous-, discrete-time,difference equations are particular cases of this definition. CHALLENGE: tocreate general sliding mode and sliding mode control theory in terms of shiftoperators embracing all types of control systems.

This author hopes that in the very beginning of the new millennium wewill have a chance to participate in discussion of the-here-mentioned problemsof VSS theory and (what is much more interesting) a set of new ones unknowntoday.

References

1. Emel’yanov, S.V. (1957). Method of designing complex control algorithms usingan error and its first time-derivative only, Automation and Remote Control,18(10) (In Russian).

2. Letov, A.M. (1957). Conditionally stable control system (on a class of optimalsystem), Automation and Remote Control, 18(7) (In Russian).

3. Emel’yanov, S.V., Burovoi I.A. and et. al. (1964). Mathematical models ofprocesses in technology and development of variable structure control systems,No.21, Metallurgy, Moscow (In Russian).

4. Emel’yanov, S.V. , Taran, V.A. (1962). On a class of variable structure controlsystems, Proc. of USSR Academy of Sciences, Energy and Automation, No.3(In Russian).

5. Barbashin, E.A. (1967). Introduction in stability theory, Nauka, Moscow (InRussian).

6. Bezvodinskaya, T.A. and Sabayev, E.F. (1974). Stability conditions in large forvariable structure systems, Automation and Remote Control, 35(10), (P.1).

7. Kostyleva, N.E. (1964). Variable structure systems for plants with zeros in atransfer function, Theory and application of control systems, Nauka, Moscow(In Russian).

8. Taran, V.A. (1964). Control of linear plant by an I-controller with variablestructure without ideal derivatives in a control low, Automation and RemoteControl, 25(11) (In Russian).

9. Taran, V.A. (1965). Design of control systems using switching phase-shiftingfilters, Proc. of USSR Academy of Sciences, Energy and Automation, No.4 (InRussian).

10. Dubrovski, E.N. (1967). Adaptation principle in variable structure systems,Proceedings of 2nd Bulgarian Conference on Control, 1, part 1, Varna (InRussian).

11. Andronov, A.A., Vitt, A.A. and Khaikin, S.E. (1959). Theory of oscillations,Fizmatgiz, Moscow (In Russian).


12. Theory of variable structure systems, S.V. Emel’yanov Ed., Nauka, Moscow,1970 (In Russian).

13. Kornilov, Yu.G. (1950). On effect of controller insensitivity on dynamics ofindirect control, Automation and Remote Control, 11(1) (In Russian).

14. Popovski, A.M. (1950). Linearization of sliding operation mode for a constantspeed controller, Automation and Remote Control, 11(3) (In Russian).

15. Dolgolenko, Yu.V. (1955). Sliding modes in relay indirect control systems, Pro-ceeding of 2nd All-Union Conference on Control, 1, Moscow (in Russian).

16. Filippov, A.F. (1961). Application of the theory of differential equations withdiscontinuous right-hand sides to non-linear problems of automatic control,Proceedings of 1st IFAC Congress II, Butterworths, London.

17. Neimark, Yu.I. (1961). Note on A. Filippov’s paper, Proceedings of 1st IFACCongress II, Butterworths, London.

18. Andre, J. and Seibert, P. (1956). Uber stuckweise lineare Differential-gluichungen bei Regelungsproblem auftreten, I,II, Arch. Der Math., 7, Nos.2 und 3.

19. Utkin, V.I. (1972). Equations of slipping regime in discontinuous systems, Au-tomation and Remote Control, 33(2).

20. Drazenovic, B. (1969). The invariance conditions in variable structure systems,Automatica, 5(3), Pergamon Press.

21. Rashevski,P.K. (1947). Geometric approach to partial differential equations,Gostechizdat,Moscow (in Russian).

22. Gutman, S. and Leitmann, G. (1976). Stabilizing feedback control for dynamicsystems with bounded uncertainties. Proceedings of IEEE Conference on De-cision and Control.

23. Pontryagin, L.S. (1955). Remark in discussion, Proceeding of 2nd All-UnionConference on Control, 1, Moscow (in Russian).

Books on VSS and Sliding Mode Control (in English)

1. V. Utkin, Sliding Modes and their Applications in Variable Structure Sys-tems, Mir,Moscow, 1978, (Translation of the book published by Nauka,Moscow, 1974 in Russian).

2. Yu. Itkis, U., Control Systems of Variable Structure, Wiley, New York,1976.

3. Deterministic Non-Linear Control, A.S. Zinober, Ed., Peter PeregrinusLimited, UK, 1990.

4. Variable Structure Control for Robotics and Aerospace Application, K-K.D. Young, Ed., Elsevier Science Publishers B.V., Amsterdam, 1993.

5. Variable Structure and Lyapunov Control, A.S.Zinober, Ed.,Springer Ver-lag, London, 1993.

6. V. Utkin, Sliding Modes in Control and Optimization, Springer Verlag,Berlin, 1992.

7. Variable Structure Systems, Sliding Mode and Nonlinear Control , K.D.Young and U. Ozguner (Eds), Springer Verlag, 1999.

8. C. Edwards and S. Spurgeon, Sliding Mode Control: Theory and Appli-cations, Taylor and Frencis, London, 1999.

9. V. Utkin, J. Guldner and J.X. Shi, Sliding Mode Control in Electro-Mechanical Systems, Taylor and Frencis, London, 1999.

32 V.I. Utkin

An Integrated Learning Variable StructureControl Method

Jian-Xin Xu

E.C.E. Dept., National University of Singapore, Singapore 117576

Subtitle: For the finite period tracking control task repeatable over iterations,the uniformly bounded learning control is synthesized into variable structurecontrol to approximate the equivalent control and to realize perfect tracking.

Abstract. In this chapter, we consider repeatable tracking control tasks using anew control approach - Learning Variable Structure Control (LVSC). LVSC synthe-sizes two main control strategies: Variable Structure Control (VSC) as the robustpart and learning control as the intelligent part. The incorporation of the powerfullearning function, by virtue of the internal model principle, completely nullifies thetracking error. The switching control mechanism on the other hand, retains the wellappreciated properties of VSC, especially the insensitivity to norm-bounded systemuncertainties. Through a rigorous proof based on the energy function and functionalanalysis, we show that the LVSC system achieves the following novel properties: (1)the tracking error sequence converges uniformly to zero; (2) the bounded learningcontrol sequence converges to the equivalent control, i.e. the desired control profilealmost everywhere; (3) the system state sequence and VSC control sequence areuniformly continuous. To address important practical considerations, the learningmechanism is implemented by means of Fourier series expansions, hence it achievesbetter tracking performance.

1 Introduction

A. Variable Structure Control (VSC) is well noted for its simplicity in im-plementation and outstanding robustness to system uncertainties when insliding mode [1–5]. A switching control with a signum function best catersto the worst case control environment where system perturbations can bestructured, unstructured, deterministic, stochastic, nonvanishing, and onlyupper bounds of system perturbations are available. Replacing the signumfunction with a continuous function will incur inferior control performancesuch as less tracking accuracy [6]. Numerous schemes have been developed toimprove tracking accuracy by incorporating VSC with other advanced controlmethods [7–10]. It should be noted that a typical VSC with discontinuousgain at the equilibrium is able to generate an infinitely high gain to suppressany bounded disturbances, e.g.

uvsc = k sign(σ)


where k is a positive gain, σ is a switching quantity, and the signum functionsign(σ) = 1 when σ > 0 and sign(σ) = −1 when σ < 0. In the worst casecontrol environment where the process is subject to non-periodic or randomperturbations, any advanced and intelligent control schemes will fail to work,and the typical VSC with infinite gain is perhaps the only control schemestill able to achieve perfect trajectory tracking and perfect disturbance re-jection. Nevertheless the chattering problem arising in real applications andimplementation prevents the use of such an infinite gain. In addition, in thesampled or discrete-time circumstances, VSC will lose its high gain propertyanyway due to the stability concern. Numerous smoothing schemes have beenproposed aiming at eliminating chattering by shaping the control gain, such asusing a continuous function to replace the signum function. All these schemes,however, have a side effect of losing tracking accuracy. The smoother the per-formance, the lower the gain around the equilibrium, and in the sequel thelower the tracking accuracy. The reason is obvious, at the equilibrium σ = 0,any continuous function of σ will be zero accordingly, nevertheless the pro-cess may still demand a non-zero control action in order to sustain the track-ing task or cancel the non-vanishing disturbances. To overcome the aboveinherent limitation of feedback, we have to consider a feedforward controltogether with feedback. Many feedforward compensation approaches are ap-plicable when the control environment is not the worst. Among them learningcontrol is one such approach particularly effective to the repeatable controlenvironment, i.e. for periodic reference trajectories and periodic disturbances.

B. There are many ways to combine VSC and LC (learning control). Herewe show two such topological structures.

VSC

Augmented Process

Plant+

InternalMode l

Fig. 1. LVSC Structure I

In Fig.1, it shows that the process is first augmented with an appropriateinternal model. Then VSC is designed to guarantee the augmented systemto be stable and robust. In ([11]) a repetitive type VSC is made possible fora class of discrete-time LTI systems. As long as the reference signal r(j) andthe disturbance d(j) is of period T which is a positive integer, the simplestinternal model is zT − 1. We can see that any error signal e(j) of period T ,

34 J.-X. Xu

once passing through the factor zT − 1, will be zero simply because

(zT − 1)e(j) = e(j + T )− e(j) = e(j)− e(j) = 0.

An immediate advantage is, such an augmented system is able to yield thedesired internal model which is the same as the reference signal generator anddisturbance generator, no matter how complicated they are. All the informa-tion needed is to know the period T a priori. In other words, the processequipped with the internal model is able to absorb the periodic residue. Sucha residue in general cannot be completely eliminated by a typical error feed-back such as a VSC with smoothing scheme, because of the limited controlgain.

When the process is highly nonlinear and/or the learning mechanism isnonlinear, it is not an easy task to directly “transplant” the internal modelinto the process. An alternative way is to insert the learning mechanism inbetween the process and the VSC, as shown in Fig.2.

VSC

LC

Plant

Fig. 2. LVSC Structure II

This leads to a modular strategy. The VSC module will be first designedto ensure the global stability and achieve the uniformly bounded tracking ac-curacy with an appropriate smoothing scheme. The learning module will beadded to further improve the tracking performance whenever the control taskrepeats. Note that we should not re-design the VSC module in this configura-tion. In case the learning module is removed, as a stand-alone controller theVSC module still keeps its normal function. The modular approach providesextra flexibility as one can easily add a learning mechanism to an existingVSC system without any modification, hence it is easy to implement.

It is worthy to highlight that, most existing repetitive control or iterativelearning control methods are of the typical feedforward class and thus sen-sitive to any non-periodic factors. By incorporating learning into VSC, theVSC part will “protect” the learning part to certain extent by virtue of itsexcellent robustness property. Two kinds of control tasks can be easily han-dled: the first with periodic reference and disturbance, and the second withrepeated reference and disturbance. The former is a kind of repetitive VSC,and the latter is a kind of iterative learning VSC. In this chapter we will

35An Integrated Learning Variable Structure Control Method

give detailed design and analysis of the latter, named as Learning VariableStructure Control (LVSC).

C. Generating the equivalent control profile is the ultimate objective of VSCin sliding mode, which assures perfect tracking and complete disturbancerejection. Under the boundedness and Lipschitz continuity conditions of thesystem dynamics, [1] provides an approach to acquire the equivalent control:passing the control signal through a first-order low-pass filter. It requires thatthe system stays strictly on the switching surface and the time constant ofthe filter approach zero. In other words, it demands an infinite switchingfrequency and an infinite filter bandwidth in order to deal with the worstcase control environment.

In a repeatable control environment, we can make use of learning con-trol approaches. The stringent requirements for obtaining equivalent controlin [1] through filtering control signals can thus be relaxed. In this chapterwe focus on learning for the deterministic control environment which hasbeen well summarized in [12], [13], etc. The non-repeatable factors such asrandom perturbations are assumed to be much smaller and consequently neg-ligible. Therefore the learning introduced here is more like iterative learningapproach instead of a stochastic learning approach such as Bayesian learningor learning automaton. Moreover, the robustness of the control system willstill be ensured to certain extent by VSC. It is worthy to point out again, thelearnability is highly related to the repeatability of the control environment.LVSC can achieve better control performance when the system repeatabilityincreases. By virtue of repeatability, the past control tracking error sequencesdo reflect the dynamic characteristics of the uncertain system, and reflectthe influence from the reference signal as well as system perturbations to thetracking errors. Learning in LVSC aims at extracting useful control knowl-edge from past control and tracking error sequences so as to approximateequivalent control.

As a modular approach, the proposed LVSC has a very simple structureconsisting of two components in additive form: a standard switching controlmechanism based on the known upper bounds using a continuous smoothingfunction, and a learning mechanism which simply adds up either a past track-ing error sequence or a past VSC sequence. Through rigorous proof based onenergy function and functional analysis, we show that LVSC system achievesthe following novel properties: (1) the tracking error sequence converges uni-formly to zero; (2) the learning control sequence which is bounded everywhereconverges to the equivalent control, i.e. the desired control profile almost ev-erywhere (a.e.); (3) the system state sequence and VSC control sequence areuniformly continuous.

In this chapter we further address important issues arising from practi-cal considerations and implement the proposed learning variable structurecontrol in the frequency domain by means of Fourier series expansion, which

36 J.-X. Xu

achieves the best functional approximation for periodic functions in the senseof the mean square. In addition, Fourier series-based learning can further en-hance the robustness property of LVSC and improve tracking performance.

2 Notation and Preliminaries

Rn denotes space of n-tuples of real numbers. |z| denotes the absolute value

of a function z. ‖v‖ =(vT v

) 12 denotes the norm of a vector v ∈ Rn. ‖A‖ =√

λmax (ATA) denotes the norm of a matrix A. Z+= 1, 2, · · · denotes the

set of positive integers. B(D) denotes a space of bounded functions on D.C(D) denotes a space of continuous functions on D. UC(D) denotes a spaceof uniformly continuous functions on D. Cn(D) denotes a space of n timescontinuously differentiable functions on D.

|h|w denotes an extended time-weighted L2 norm of function h defined by

|h|w =

∫ t

0e−λτh2(τ)dτ for a positive constant λ < ∞. h and 0 are said to be

equivalent if |h|w = 0, or h = 0 almost everywhere [14].Holder inequality [15] If p, q ∈ [1,∞] and p−1+ q−1 = 1, then f ∈ Lp, g ∈

Lq imply that fg ∈ L1 and ‖fg‖1 ≤ ‖f‖p‖g‖q. When p = q = 2, the Holderinequality becomes the Schwartz inequality, i.e., ‖fg‖1 ≤ ‖f‖2‖g‖2.

Bellman-Gronwall Lemma I [15] Let λ(t), g(t), k(t) be nonnegativepiecewise continuous functions of time t. If the function y(t) satisfies theinequality y(t) ≤ λ(t) + g(t)

∫ t

t0k(τ)y(τ)dτ, ∀t ≥ t0 ≥ 0, then y(t) ≤ λ(t) +

g(t)∫ t

t0λ(s)k(s)e

∫ t

sk(τ)g(τ)dτ

ds, ∀t ≥ t0 ≥ 0.Bellman-Gronwall Lemma II [15] Let λ(t), k(t) be nonnegative piecewise

continuous function of time t and let λ(t) be differentiable. If the functiony(t) satisfies the inequality y(t) ≤ λ(t) +

∫ t

t0k(s)y(s)ds, ∀t ≥ t0 ≥ 0, then

y(t) ≤ λ(t0)e∫ t

t0k(s)ds

+∫ t

t0λ(s)e

∫ t

sk(τ)dτ

ds, ∀t ≥ t0 ≥ 0.Continuity Theorem [16] Let f ∈ B(D)∩C(D). Suppose φ(t) is a solution

of x = f(x, t) where x ∈ Rn and t ∈ R1 ∀t ∈ (a, b). Then if (a, φ(a+)) and(b, φ(b−)) are in D where φ(a+) = limt→a+ φ(t) and φ(b−) = limt→b− φ(t),then the solution φ(t) ∈ C[a, b].

Cantor Theorem [17] Let f ∈ C(D), if D is a closed nonempty set, thenf ∈ UC(D).

3 Problem Formulation and LVSC Configuration

Consider the nth-order deterministic nonlinear uncertain dynamical systemdescribed by

xj = xj+1

xn = f(x, t) + b(x, t)u . (1)


x = [x1, · · ·, xn]T ∈ X ⊆ Rn is the physically measurable state vector. u isthe control input. f(x, t) and b(x, t) are nonlinear uncertain functions.

A: Repeatable tracking control tasksGiven a finite initial state xi(0) and a finite time interval [0, Tf ] where

i denotes the iteration sequence, the control objective is to design a VSCcombined with iterative learning such that, as i → ∞, the system state xi

of the nonlinear uncertain system (1) tracks the desired trajectory xd =[xd,1 · · · xd,n]T ∈ XD ⊆ Rn where xd ∈ C1([0, Tf ]) and is generated by thefollowing dynamics over [0, Tf ]

xd,j = xd,j+1

xd,n = β(xd, t) + r(t) (2)

where β(xd, t) ∈ C(XD× [0, Tf ]) is a known function and r(t) ∈ C([0, Tf ]) isa reference input. xd ∈ C1([0, Tf ]) ensures xd ∈ C([0, Tf ]), xd ∈ C([0, Tf ])and therefore xd ∈ B([0, Tf ]) and xd ∈ B([0, Tf ]) as [0, Tf ] is a finiteinterval. As part of the repeatability condition, the initial states xi(0) =xd(0), i.e., σi(0) = 0 are available for all trials.

B: Switching surfaceThe switching surface of a basic VSC is defined as

σi = (xd,n − xi,n) +n−1∑j=1

αj (xd,j − xi,j) = αT (xd − xi) (3)

where αT =[αT

1 , 1], αT

1 = [α1, · · · , αn−1] and αj (j = 1, · · · , n − 1) arechosen coefficients such that the polynomial sn−1 +αn−1s

n−2 +αn−2sn−3 +

· · ·+ α1 is Hurwitz and s is a Laplace operator.C: Assumption

(A1) f(x, t) is bounded by a known bounding function fmax(x, t), i.e.,|f(x, t)| ≤ fmax(x, t) while b(x, t) is positive definite which is lower boundedby a known bounding function bmin(x, t) and is also upper bounded, i.e., 0 <b ≤ bmin(x, t) ≤ b(x, t) ≤ b for some positive constants b, b and ∀(x, t) ∈ X×[0, Tf ]. It implies that b−1(x, t)f(x, t) is also bounded as |b−1(x, t)f(x, t)| ≤b−1min(x, t)fmax(x, t). The bounding functions fmax(x, t) and bmin(x, t) belongto B (X × [0, Tf ]) ∩ C (X × [0, Tf ]).

(A2) ∀h ∈ f, b, h(x, t) ∈ C (X × [0, Tf ]) and h(x, t) satisfies the glob-ally Lipschitz condition, ‖h(x1, t)−h(x2, t)‖ ≤ lh‖x1(t)−x2(t)‖, ∀t ∈ [0, Tf ],∀x1,x2 ∈ X and for a positive constant lh < ∞.

D: Equivalent control ueq

Differentiating (3) with respect to time results

σi = (xd,n − xi,n) +n−1∑j=1

αj (xd,j+1 − xi,j+1) . (4)

38 J.-X. Xu

Substituting (1) and (2) into (4) gives

σi =n−1∑j=1

αj (xd,j+1 − xi,j+1) + β(xd, t) + r(t)− fi − biui

= gi − fi − biui (5)

where fi= f(xi, t), bi

= b(xi, t), gi

=

∑n−1j=1 αj (xd,j+1 − xi,j+1) + gd, gd

=

β(xd, t) + r(t). According to (5), making σi = 0 obtains σi(t) = σi(0) = 0,i.e., xi(t) = xd(t) ∀t ∈ [0, Tf ] and gives the following expression for theequivalent control

ueq = b−1d gd − b−1

d fd (6)

where bd= b(xd, t), fd

= f(xd, t). Since xd ∈ B([0, Tf ]), then gd ∈ B([0, Tf ])

as β(xd, t) ∈ C(XD×[0, Tf ]), r(t) ∈ C([0, Tf ]) and [0, Tf ] is a finite interval.According to A1, b−1

d ≤ b−1 and xd ∈ B([0, Tf ]) ensures fd ∈ B([0, Tf ]),hence ueq ∈ B([0, Tf ]).

E: LVSC configurationThe proposed LVSC is given below

ui = uv,i + u∗sat (ul,i−1, u∗) , ul,0 = 0, (7)

uv,i = ζσi + ρisat(σi, ε), ρi = b−1min(xi, t) [|gi|+ fmax(xi, t)] , (8)

ul,i = u∗sat (ul,i−1, u∗) + ιi, ιi ∈ βlσi, uv,i , (9)

sat(1, ∗) =

1/∗ | 1 | ≤ ∗

sgn(1) | 1 | > ∗ (10)

where ζ, ε, βl and u∗ are positive-definite constants. u∗ is a sufficiently largeconstant such that

u∗ ≥ supt∈[0, Tf ]

|ueq(t)| (11)

to ensure learnability. Note that u∗ can be either decided from the real lim-itation of the physical process, or simply chosen to be an arbitrarily largebut finite virtual bound which does not affect the VSC part. (8) is a VSClaw which replaces the signum function sgn(σi) with the saturation functionsat(σi, ε) to eliminate chattering [6]. Since uv,i is continuous with respect toσi, gi, bmin(xi, t) and fmax(xi, t), bmin(xi, t) ∈ C(X × [0, Tf ]), fmax(xi, t) ∈C(X × [0, Tf ]) according to A1, σi ∈ C(X × [0, Tf ]), gi ∈ C(X × [0, Tf ]) asxd ∈ C([0, Tf ]), hence uv,i ∈ C (X × [0, Tf ]).

According to (10), the VSC law (8) is equivalent to

uv,i = (ζ + ρiki)σi, ki=

1/ε |σi| ≤ ε1/|σi| |σi| > ε

. (12)

The above definition ensures that ki ≤ 1/ε.


Since bd = 0, it can be shown by (6) that, if β(xd, t)+ r(t)− f(xd, t) = 0,then ueq = 0. It is impossible to generate ueq by using the VSC law (8),since when xi → xd, σi → 0, then uv,i → 0 = ueq. To produce ueq or toachieve perfect tracking using a smooth control input, (9) introduces thesimple structure of the proposed learning control ul,i where the item ιi isfor updating learning control which will be discussed in detail in subsequentsections.

To evaluate the learning performance, the following time-weighted L2

norm of ul,i − ueq is used

Ji(t) = |ul,i − ueq|w =∫ t

0

e−λτ [ul,i(τ)− ueq(τ)]2dτ ≥ 0 (13)

Under saturator (10) and condition (11), the following key inequality holds

[u∗sat (ul,i−1, u∗)− ueq]

2 ≤ (ul,i−1 − ueq)2, ∀i ∈ Z+ ∩ (i ≥ 2).

From (9) and the above, the difference of Ji(t) between two successive trials,∀i ∈ Z+ ∩ (i ≥ 2) can be derived as

∆Ji(t) = Ji(t)− Ji−1(t)

=∫ t

0

e−λτ[(ul,i − ueq)

2 − (ul,i−1 − ueq)2]dτ

≤∫ t

0

e−λτ(ul,i − ueq)

2 − [u∗sat (ul,i−1, u∗)− ueq]

2dτ

=∫ t

0

e−λτ [ul,i − u∗sat(ul,i−1, u∗)] [ul,i + u∗sat(ul,i−1, u

∗)− 2ueq] dτ

=∫ t

0

e−λτι2i + 2ιi [u∗sat(ul,i−1, u

∗)− ueq]dτ. (14)

From (5), it can be obtained that

b−1i σi = b−1

i gi − b−1i fi − ui.

Substituting the control law (7) into the above yields

b−1i σi = b−1

i gi − b−1i fi − uv,i − u∗sat (ul,i−1, u

∗) .

Subtracting ueq in (6) from both sides of the above yields

b−1i σi − ueq =

(b−1i gi − b−1

i fi

)− (

b−1d gd − b−1

d fd

) − uv,i − u∗sat (ul,i−1, u∗)

which can be rewritten into

u∗sat (ul,i−1, u∗)− ueq = −uv,i − b−1

i σi − γi, (15)

where

γi = (b−1d fd − b−1

i fi)− (b−1d gd − b−1

i gi). (16)

40 J.-X. Xu

It can be derived that

|γi| ≤ |b−1d fd − b−1

i fd + b−1i fd − b−1

i fi|+|b−1

d gd − b−1i gd + b−1

i gd − b−1i gi|

≤ b−1i b−1

d |bd − bi| · |fd|+ b−1i |fd − fi|

+b−1i b−1

d |bd − bi| · |gd|+ b−1i |gd − gi|.

Since gd − gi = −∑n−1j=1 αj (xd,j+1 − xi,j+1), we have

|gd − gi| ≤(α2

1 + · · ·+ α2n−1

) 12 ‖xd − xi‖.

Under assumption A1, b−1i and b−1

d are all bounded by b−1. Since fd ∈B([0, Tf ]) and gd ∈ B([0, Tf ]), we denote that fd = supt∈[0, Tf ] fd(t) andgd = supt∈[0, Tf ] gd(t). Using the globally Lipschitz condition described in A2we can obtain

|γi| ≤ c‖xd − xi‖ (17)

where c = b−1[lbb

−1fd + lf + b−1lbgd +(α2

1 + · · ·+ α2n−1

) 12]is a finite pos-

itive constant. From (9) and (15) it can be obtained that

ul,i − ueq = ιi − uv,i − b−1i σi − γi. (18)

F. Property AnalysisTo facilitate LVSC analysis, here we give four propositions which reveal

the boundedness relationship among the quantities σi, σi, xi, xi, γi, ρi, uv,i,ul,i, ui and Ji.

Proposition 1. For system (1) given the desired trajectory in (2) and switch-ing surface (3), the following hold

xd − xi = A (xd − xi) + bσi, (19)

‖xd − xi‖ ≤ ‖A‖∫ t

0

|σi(τ)|e‖A‖(t−τ)dτ + |σi| (20)

where A =[0(n−1)×1 I(n−1)×(n−1)

0 −αT1

], b =

[01×(n−1) 1

]T .

Proof. Combining (1) and (2) yields

xd,j − xi,j = xd,j+1 − xi,j+1. (21)

Rearranging (4) gives

xd,n − xi,n = −n−1∑j=1

αj (xd,j+1 − xi,j+1) + σi. (22)


Combining (21) and (22) gives (19). Integrating both sides of (19) and notic-ing σi(0) = 0 and xi(0) = xd(0) one obtains

xd − xi = A

∫ t

0

(xd − xi) dτ + bσi.

Taking the norm of the above and since ‖b‖ = 1, the following stands

‖xd − xi‖ ≤ ‖A‖∫ t

0

‖xd − xi‖dτ + |σi|.

Applying the Bellman-Gronwall Lemma I, we can obtain (20).

Proposition 2. For system (1) satisfying assumptions A1, under the controllaws (7)-(10), σi ∈ B([0, Tf ]), xi ∈ B([0, Tf ]), σi ∈, xi ∈ B([0, Tf ]),ρi ∈ B([0, Tf ]), i.e., ρi ≤ ρ for a positive constant ρ < ∞, uv,i ∈ B([0, Tf ]),ul,i ∈ B([0, Tf ]), ui ∈ B([0, Tf ]) and Ji ∈ B([0, Tf ]), ∀t ∈ [0, Tf ] and∀i ∈ Z+.

Proof. Define a Lyapunov function as Vi = σ2i /2. Differentiating Vi with

respect to t using (5) one obtains

Vi = −biσi

[ui −

(b−1i gi − b−1

i fi

)].

If |σi| ≥ ε, under (10) and assumptions A1, substituting control laws (7), (8)into the above yields

Vi = −biζσ2i − biρi|σi|+ biσi

(b−1i gi − b−1

i fi

) − biσiu∗sat (ul,i−1, u

∗)

≤ −biζσ2i − bi|σi|

[ρi −

(b−1i gi − b−1

i fi

)sgn(σi)

]+ biu

∗|σi|≤ −biζσ

2i + biu

∗|σi|.Since σi(0) = 0, ∀i ∈ Z+, it can be concluded that system stays inside thebound |σi(t)| ≤ max

ε, ζ−1u∗, ∀t ∈ [0, Tf ].

According to (20) of Proposition 1, σi ∈ B([0, Tf ]) ensures that xi ∈B([0, Tf ]) since xd ∈ B([0, Tf ]). From A1, fmax(xi, t) and bmin(xi, t) belongto B(X× [0, Tf ]) ensures that ρi ∈ B(X× [0, Tf ]) as xd ∈ B([0, Tf ]), hencexi ∈ B([0, Tf ]) brings that ρi ∈ B([0, Tf ]) according to (8). ρi ∈ B([0, Tf ]),ki ≤ 1/ε and σi ∈ B([0, Tf ]) lead to uv,i ∈ B([0, Tf ]) according to (12).Since u∗sat (ul,i−1, u

∗) ≤ u∗, then σi ∈ B([0, Tf ]) and uv,i ∈ B([0, Tf ]) leadto ui ∈ B([0, Tf ]) and ul,i ∈ B([0, Tf ]) according to (7) and (9), hencexi ∈ B([0, Tf ]) according to (1) and σi ∈ B([0, Tf ]) according to (5). From(13), ul,i ∈ B([0, Tf ]) ensures Ji ∈ B([0, Tf ]), ∀i ∈ Z+.

Proposition 3. For system (1) given the desired trajectory in (2) and switch-ing surface (3), under assumption A1 and A2, the following stands∫ t

0

e−λτ |σi(τ)| · |γi(τ)|dτ ≤(c+ c‖A‖Tfe

‖A‖Tf

) ∫ t

0

e−λτσ2i (τ)dτ. (23)

42 J.-X. Xu

Proof. Since 0 ≤ ν ≤ τ ≤ t ≤ Tf , then 0 ≤ τ − ν ≤ τ ≤ Tf and −λ2 τ ≤ −λ

2 ν.Using the Holder inequality, it can be obtained from (17) and (20) underassumption A2 that∫ t

0

e−λτ |σi(τ)| · |γi(τ)|dτ − c

∫ t

0

e−λτσ2i (τ)dτ

≤ c‖A‖∫ t

0

e−λτ |σi(τ)|[∫ τ

0

e‖A‖(τ−ν)|σi(ν)|dν]dτ

≤ c‖A‖e‖A‖Tf

∫ t

0

e−λ2 τ |σi(τ)|

[∫ τ

0

e−λ2 τ |σi(ν)|dν

]dτ


[∫ t

0

e−λ2 τ |σi(τ)|dτ

]2


[∫ t

0

e−λτσ2i (τ)dτ

] [∫ t

0

12dτ

]

≤ c‖A‖Tfe‖A‖Tf

∫ t

0

e−λτσ2i (τ)dτ.

Rearranging the above gives (23).

Since ιi ∈ βlσi, uv,i is used in the learning law (9), According to (12),we have

ιi − uv,i = κiσi (24)

where κi=

βl − ζ − ρiki ιi = βlσi

0 ιi = uv,i. Since ρi ≤ ρ according to Proposition

2 under assumption A1 and ki ≤ 1/ε, we have |κi| ≤ κ for a positive constantκ < ∞.

Proposition 4. For system (1) given the desired trajectory in (2) and switch-ing surface (3), under assumptions A1 and A2, if the control laws (7), (8)and learning law (9) are applied, then the following stands

‖xi − xd‖ ≤ belTfT12

f J12i (Tf ), (25)

|σi| ≤ b(α2

1 + · · ·+ α2n−1 + 1

) 12 elTfT

12

f J12i (Tf ). (26)

where l1 = ‖A‖+ bc+ bκ(α21 + · · ·+ α2

n−1 + 1)12 and l

= max(λ, l1).

Proof. From (18) it can be obtained that

σi = bi (ιi − uv,i)− biγi − bi (ul,i − ueq) .

Substituting the above into (19) yields

xd − xi = A (xd − xi) + b [bi (ιi − uv,i)− biγi − bi (ul,i − ueq)] .


Since xi(0) = xd(0), ‖b‖ = 1, bi ≤ b, according to (24) and |κi| ≤ κ underassumption A1, it can be obtained from the above that

‖xd − xi‖ ≤ ‖A‖∫ t

0

‖xd − xi‖dτ

+b

∫ t

0

|ul,i − ueq|dτ + bκ

∫ t

0

|σi|dτ + b

∫ t

0

|γi|dτ.

Under the assumptions A1 and A2, substituting (17) into the above yields

‖xd − xi‖ ≤ (‖A‖+ bc) ∫ t

0

‖xd − xi‖dτ

+b

∫ t

0

|ul,i − ueq|dτ + bκ

∫ t

0

|σi|dτ. (27)

From (3) we have

|σi| ≤(α2

1 + · · ·+ α2n−1 + 1

) 12 ‖xd − xi‖ (28)

and from which it can be obtained by the Holder inequality and Bellman-Gronwall Lemma II that

‖xd − xi‖ ≤ l1

∫ t

0

‖xd − xi‖dτ + b

∫ t

0

|ul,i − ueq|dτ

≤∫ t

0

bel(t−τ)|ul,i − ueq|dτ ≤ belTf

∫ Tf

0

e−lτ |ul,i − ueq|dτ

≤ belTf

[∫ Tf

0

e−2lτ (ul,i − ueq)2dτ

] 12

[∫ Tf

0

12dτ

] 12

≤ belTfT12

f

[∫ Tf

0

e−λτ (ul,i − ueq)2dτ

] 12

= belTfT12

f J12i (Tf ).

Hence from (28) and the above, we can obtain (26) which completes theproof.

4 LVSC with σi-Updating

In this section, we consider the learning law (9) with ιi = βlσi, i.e.,

ul,i = u∗sat(ul,i−1, u∗) + βlσi (29)

where βl is a positive learning gain. Note that this learning updating law,when working in an unsaturated region, is analogous to most P type iterativelearning control algorithms. Here the key point is, the above learning law canwork concurrently with the VSC law to achieve robust learning control.

44 J.-X. Xu

Theorem 1. Consider the nonlinear system (1) satisfying assumptions A1and A2 and giving a desired trajectory xd defined by (2). Under the controllaws (7), (8) and learning law (29), as i → ∞, σi(t) converges uniformly to 0,xi(t) converges uniformly to xd(t) and ul,i(t) converges to ueq(t), ∀t ∈ [0, Tf ]almost everywhere.

Proof. Substituting (15) into (14) under ιi = βiσi obtains the difference ofJi(t) between two successive trials ∀i ∈ Z+ ∩ (i ≥ 2) as

∆Ji(t)= Ji(t)− Ji−1(t)

=∫ t

0

e−λτ(β2

l σ2i − 2βluv,iσi − 2βlb

−1i σiσi − 2βlσiγi

)dτ.

From (12), it can be found that uv,iσi ≥ 0, then

∆Ji(t) ≤∫ t

0

e−λτ(β2

l σ2i − 2βlb

−1i σiσi − 2βlσiγi

)dτ

≤ −2βl

∫ t

0

b−1i e−λτσiσidτ +

∫ t

0

e−λτ(β2

l σ2i + 2βl|σi| · |γi|

)dτ

= −βl

∫ σ2i (t)

0

b−1i e−λτdσ2

i (τ) +∫ t

0

e−λτ(β2


)dτ.

Since σi(0) = 0, using A1, Proposition 3 and taking the integration by partsone obtains

∆Ji(t) ≤ −βlb−1

∫ σ2i (t)

0

e−λτdσ2i (τ) +

∫ t

0

e−λτ(β2


)dτ

= −βlb−1e−λtσ2

i (t)

−λβlb−1

∫ t

0

e−λτσ2i dτ +

∫ t

0

e−λτ(β2


)dτ

≤ −βlb−1e−λtσ2

i (t)

−βlb−1

[λ−

(βl + 2c+ 2c‖A‖Tfe

‖A‖Tf

)b] ∫ t

0

e−λτσ2i (τ)dτ.

Since(βl + 2c+ 2c‖A‖Tfe

‖A‖Tf)b is a finite positive constant, there exists a

sufficiently large λ such that

λ ≥(βl + 2c+ 2c‖A‖Tfe

‖A‖Tf

)b (30)

to ensure that

∆Ji(t) ≤ −βlb−1e−λtσ2

i (t). (31)

From the above we can see that too small a learning gain βl slows downthe decrease of Ji(t). However, a larger βl needs a larger λ to ensure theconvergence as shown in (30), and also a smaller e−λt. Moreover, a smaller


e−λt underestimates the learning convergence ul,i(t) → ueq(t) evaluated by(13) more. Hence a moderate βl should be used to decrease Ji(t) faster, andto speed up the learning convergence.

According to (13), Ji(t) ≥ 0, then from (31) we have 0 ≤ Ji(t) ≤ Ji−1(t) ≤··· ≤ J1(t). From (31), taking the summation of ∆Jj(t) over j = 1 to i obtains

Ji(t)− J1(t) ≤ −βlb−1e−λt

i∑j=1

σ2j (t).

As Ji ≥ 0, we have from the above that limi→∞∑i

j=1 σ2j (t) ≤ β−1

l beλtJ1(t)which concludes that limi→∞ σi(t) = 0, ∀t ∈ [0, Tf ] due to that J1(t) ∈B([0, Tf ]).

Substituting ιi = βlσi into (18) one obtains

ul,i − ueq = βlσi − uv,i − b−1i σi − γi. (32)

According to (20), limi→∞ σi = 0 leads to limi→∞ xi = xd which brings thatlimi→∞ uv,i = 0 and limi→∞ γi = 0 according to (8) and (17). Thus it canbe obtained from (13) and (32) that

limi→∞

Ji(Tf ) = limi→∞

∫ Tf

0

e−λτ [ul,i(τ)− ueq(τ)]2dτ

= limi→∞

∫ Tf

0

e−λτ[βlσi − uv,i − b−1

i σi − γi

]2dτ

= limi→∞

∫ Tf

0

e−λτ b−2i σ2

i dτ

= limi→∞

∫ σi(Tf )

0

e−λτ b−2i σidσi(τ).

Since e−λt ≤ 1, b−2i ≤ b−2 according to A1 and σi ∈ B([0, Tf ]) according to

Proposition 2, limi→∞ σi(Tf ) = 0 concludes that

limi→∞

Ji(Tf ) = 0 (33)

and ul,i(t) converges to ueq(t) almost everywhere, ∀t ∈ [0, Tf ]. Using Propo-sition 4 and (33), we have

limi→∞

supt∈[0, Tf ]

|σi(t)| = 0, limi→∞

supt∈[0, Tf ]

‖xd(t)− xi(t)‖ = 0,

i.e., σi and xi are uniformly convergent which completes the proof.

Corollary 1. Under the same conditions as those in Theorem 1, the learn-ing law (9) warrants that σi ∈ UC([0, Tf ]), xi ∈ UC([0, Tf ]) and uv,i ∈UC([0, Tf ]), ∀i ∈ Z+.

46 J.-X. Xu

Proof. Since limi→∞ Ji(Tf ) = 0 and xd ∈ UC([0, Tf ]), then ∀ε > 0, ∃N(ε) ∈Z+ and ∃δ(ε) > 0, such that |Ji(Tf )| < ε and ‖xd(t2)−xd(t1)‖ < ε, wheneveri ≥ N(ε) and t1, t2 ∈ [0, Tf ]∩[|t1 − t2| < δ(ε)]. Hence from (25) of Proposition4, ∀ε > 0, ∃N(ε) ∈ Z+ and ∃δ(ε) > 0, such that

‖xi(t1)− xi(t2)‖ = ‖xi(t1)− xd(t1) + xd(t1)− xd(t2) + xd(t2)− xi(t2)‖≤ ‖xd(t1)− xi(t1)‖

+‖xd(t2)− xd(t1)‖+ ‖xd(t2)− xi(t2)‖< 2belTfT

12

f J12i (Tf ) + ε

whenever i ≥ N(ε) and t1, t2 ∈ [0, Tf ]∩(|t1 − t2| < δ), then xi ∈ UC([0, Tf ]),∀i ≥ N(ε). Since σi ∈ C (X × [0, Tf ]), uv,i ∈ C (X × [0, Tf ]), then xi ∈UC([0, Tf ]) ensures that σi ∈ UC([0, Tf ]) and uv,i ∈ UC([0, Tf ]), ∀i ≥N(ε).

When i < N(ε), we prove σi ∈ UC([0, Tf ]), xi ∈ UC([0, Tf ]) anduv,i ∈ UC([0, Tf ]) by induction. Combining (4) and (5) obtains xd,n− xi,n =gd − fi − biui. Since xd,j − xi,j = xd,j+1 − xi,j+1, j = 1, · · ·, n− 1, we have

xd − xi = A1 (xd − xi) + b (gd − fi − biui) (34)

where A1=

[0(n−1)×1 I(n−1)×(n−1)

0 01×(n−1)

]. Under the control law (7), (34) can be

rearranged into the following

xi = χi + bbiu∗sat(ul,i−1, u

∗). (35)

where χi = A1xi + xd −A1xd − bgd + bfi + bbiuv,i.According to assumption A1, xi ∈ B([0, Tf ]) ensures that fi ∈ B([0, Tf ]).

Hence the right hand side of (35) belongs to B([0, Tf ]) as xd, xd, xi, σi, uv,i

belong to B([0, Tf ]), ∀i ∈ Z+ according to Proposition 2.Since xd ∈ C1([0, Tf ]), xd ∈ C([0, Tf ]), gd ∈ C([0, Tf ]),

uv,i ∈ C (X × [0, Tf ]) and fi ∈ C (X × [0, Tf ]), bi ∈ C (X × [0, Tf ]) ac-cording to assumption A2, then χi ∈ C(X × [0, Tf ]).

During the first iteration, since ul,0 = 0, the right hand side of (35)belongs to B([0, Tf ]) ∩ C(X × [0, Tf ]). As x1 ∈ B([0, Tf ]), then accord-ing to Continuity Theorem, x1 ∈ C([0, Tf ]), hence σ1 ∈ C([0, Tf ]) andul,1 = uv,1 ∈ C([0, Tf ]). Since sat(1, ∗) is continuous with respect to (1),hence u∗sat(ul,1, u

∗) ∈ C([0, Tf ]) which brings that x2 ∈ C([0, Tf ]), σ2 ∈C([0, Tf ]), uv,2 ∈ C([0, Tf ]) and ul,2 = u∗sat(ul,1, u

∗) + βlσ2 ∈ C([0, Tf ])owning to the similar arguments.

In general, suppose σ1, · · ·, σi−1 belong to C([0, Tf ]), hence x1, · · ·, xi−1

and uv,1, · · ·, uv,i−1 belong to C([0, Tf ]). From (9),

ul,i−1 = βlσi−1 + u∗sat(ul,i−2, u∗)

= βlσi−1 + u∗sat(βlσi−2 + u∗sat(ul,i−3, u∗), u∗)

= · · ·= βlσi−1 + u∗sat(βlσi−2

+u∗sat(βlσi−3 + · · ·+ u∗sat(βlσ1, u∗) · ··, u∗), u∗) (36)


belongs to C([0, Tf ]). Then u∗sat(ul,i−1, u∗) ∈ C([0, Tf ]) and the right hand

side of (35) belongs to B([0, Tf ])∩C(X×[0, Tf ]). According to the ContinuityTheorem, xi ∈ B([0, Tf ]) leads to xi ∈ C([0, Tf ]), hence σi ∈ C([0, Tf ])and uv,i ∈ C([0, Tf ]). As [0, Tf ] is a closed interval, according to the CantorTheorem, σi ∈ UC([0, Tf ]), xi ∈ UC([0, Tf ]) and uv,i ∈ UC([0, Tf ]),∀i < N(ε). Combining this conclusion with xi(t) ∈ UC([0, Tf ]), σi(t) ∈UC([0, Tf ]) and uv,i(t) ∈ UC([0, Tf ]), ∀i ≥ N(ε) completes the proof.

5 LVSC with uv,i-Updating

Generally speaking, the VSC part uv,i may better reflect the demand from thetracking control task in comparison with a simple feedback by βlσi. Thereforein this section we consider learning with uv,i-updating, namely in the learninglaw (9) ιi = uv,i is used

ul,i = u∗sat(ul,i−1, u∗) + uv,i. (37)

Theorem 2. Consider the nonlinear system (1) satisfying assumptions A1,A2 and giving a desired trajectory xd defined by (2). Under the control laws(7), (8) and learning law (37), σi ∈ UC([0, Tf ]), xi ∈ UC([0, Tf ]), uv,i ∈UC([0, Tf ]), ∀i ∈ Z+. As i → ∞, σi(t) converges uniformly to 0, xi(t)converges uniformly to xd(t) and ul,i(t) converges to ueq(t), ∀t ∈ [0, Tf ]almost everywhere.

Proof. Substituting (15) into (14) using ιi = uv,i and (12), the difference ofJi(t) between two successive trials ∀i ∈ Z+ ∩ (i ≥ 2) is

∆Ji(t) =∫ t

0

e−λτ[u2

v,i − 2uv,i

(uv,i + b−1

i σi + γi

)]dτ

=∫ t

0

e−λτ(−u2

v,i − 2b−1i uv,iσi − 2uv,iγi

)dτ

≤ −2∫ t

0

e−λτ b−1i uv,iσidτ − 2

∫ t

0

e−λτuv,iγidτ

= −2∫ t

0

e−λτ b−1i (ζ + ρiki)σiσidτ − 2

∫ t

0

e−λτ (ζ + ρiki)σiγidτ.

Since σi(0) = 0, ρi ≤ ρ and ki ≤ 1/ε, using A1, Proposition 3 and taking theintegration by parts one obtains

∆Ji(t) ≤ −b−1ζ

∫ σ2i (t)

0

e−λτdσ2i (τ) + 2 (ζ + ρ/ε)

∫ t

0

e−λτ |σi| · |γi|dτ

= −b−1ζe−λtσ2i (t)

−λb−1ζ

∫ t

0

e−λτσ2i dτ + 2 (ζ + ρ/ε)

∫ t

0

e−λτ |σi| · |γi|dτ

48 J.-X. Xu

≤ −b−1ζe−λtσ2i (t)

−b−1ζ[λ− 2(ζ + ρ/ε)

(c+ c‖A‖Tfe

‖A‖Tf

)bζ−1

] ∫ t

0

e−λτσ2i dτ.

Hence there exists a sufficiently large λ such that

λ ≥ 2(ζ + ρ/ε)(c+ c‖A‖Tfe

‖A‖Tf

)bζ−1

to ensure that ∆Ji(t) ≤ −b−1ζe−λtσ2i (t). Following the same procedure in

the proof of Theorem 1, we have limi→∞σi = 0. Substituting ιi = uv,i into(18) yields ul,i−ueq = −b−1

i σi−γi. As limi→∞ σi = 0, we have limi→∞ γi = 0according to (16) and (20). Thus it can be obtained from (13) that

limi→∞

Ji(Tf ) = limi→∞

∫ Tf

0

e−λτ [ul,i(τ)− ueq(τ)]2dτ

= limi→∞

∫ Tf

0

e−λτ b−2i σ2

i dτ.

Following the same procedure in the proof of Theorem 1, limi→∞ Ji(Tf ) = 0,ul,i(t) converges to ueq almost everywhere, σi(t) converges uniformly to 0 andxi(t) converges uniformly to xd(t), ∀t ∈ [0, Tf ].

Under the learning law (37), (36) is changed to be the following form:

ul,i−1 = uv,i−1 + u∗sat(ul,i−2, u∗)

= uv,i−1 + u∗sat(uv,i−2 + u∗sat(ul,i−3, u∗), u∗)

= · · ·= uv,i−1

+u∗sat(uv,i−2 + u∗sat(uv,i−3 + · · ·+ u∗sat(uv,1, u∗) · ··, u∗), u∗).

Following the same procedure in the proof of Corollary 1, we can prove thatσi ∈ UC([0, Tf ]), xi ∈ UC([0, Tf ]) and uv,i ∈ UC([0, Tf ]) which ends theproof.

Remark 1. The only direct treatment for obtaining equivalent control appearsin [1] which uses a first-order low-pass filter. It requires ∆/τ → 0 and τ → 0where |σ| ≤ ∆ and τ is the time constant of the filter. This shows the diffi-culty in achieving equivalent control in general: it demands infinite switchingfrequency to produce an infinitesimal bound ∆ of σ and an infinite filterbandwidth due to the worst case control environment. These two stringentrequirements can be relaxed where repeatable control tasks are concerned.

Remark 2. From practical point of view, a potential problem withuv,i-updating is that uv,i is in general less smooth than βlσi. Due to the “inte-gral” nature of learning updating, high frequency components in ui may growup. Fourier series based iterative learning can well overcome this problem.


6 Fourier Series Based Iterative Learning

In this section we further address important issues arising from practical con-siderations and implement learning control in the frequency domain by meansof the Fourier series expansion. First, it should be noted that the repeatabilityof the control environment implies only countable integer frequencies beinginvolved in the equivalent control profile, this ensures the implementabilityof constructing componentwise learning in the frequency domain. Second,nowadays all advanced control approaches (including LVSC) have to be im-plemented using microprocessing technology. According to sampling theory,LVSC needs only to learn a finite number of frequencies limited by one halfof the sampling frequency ωs = 2π/Ts where Ts is the sampling interval.Any attempt to learn and manipulate frequencies above that limit will becompletely meaningless. Third, most real systems can be characterized aslow-pass filter because their bandwidth is much lower than the sampling fre-quency. It is sufficient for LVSC to take into account only a small portion ofthe Fourier series in such cases.

Therefore, the spectrum of ueq, σi, uv,i, ul,i and u∗sat(ul,i−1, u∗) is located

in a relatively low frequency band compared to ωs. Assume that system (1)has a limited bandwidth [0, wb] rad/sec where ωb < ωs/2, ueq, σi, uv,i,ul,i and u∗sat(ul,i−1, u

∗) can be expressed in the following truncated FourierLinear Combiners form

ueq = ψT ν, σi = ψT ηi, uv,i = ψT θi, (38)

ul,i = ψT νi, u∗sat(ul,i−1, u∗) = ψT νi−1 (39)

where N ≥ ωb/ω, ω = 2π/Tf ,

ψ = [0.5 cosωt cos 2ωt · · · cosNωt sinωt sin 2ωt · · · sinNωt]T ,

ψ1 = [1 cosωt cos 2ωt · · · cosNωt sinωt sin 2ωt · · · sinNωt]T ,

and

ηi =2Tf

∫ Tf

0

σiψ1dτ, θi =2Tf

∫ Tf

0

uv,iψ1dτ, (40)

νi−1 =2Tf

∫ Tf

0

u∗sat(ul,i−1, u∗)ψ1dτ. (41)

Substituting (39) into the learning law with σi-updating (29), we have ψT νi =ψT (νi−1 + βlηi). Since the basis of the Fourier space is orthogonal, we havethe following frequency domain learning law with σi-updating

νi = νi−1 + βlηi. (42)

Similarly, the frequency domain learning law with uv,i-updating can be ob-tained from (37) and (39) as

νi = νi−1 + θi. (43)

50 J.-X. Xu

Since ueq is invariant over every iteration, each element of ν is constant. Thefrequency domain learning laws (42) and (43) produce estimate νi whichconverges to ν as i → ∞.

The main advantage of the Fourier series based learning is the enhance-ment of LVSC robustness and the improvement of tracking performance. Notethat there always exists system noise or other small non-repeatable factorseven in a repeatability dominant control environment. Accumulation of thesetiny components contained in control sequence uv,i and tracking error se-quence σi may degrade the approximation precision of the learning controlsequence ul,i. The Fourier series based learning mechanism, on the otherhand, updates coefficients νi, νi−1, ηi and θi of the learned frequency com-ponents ψ and those coefficients are calculated according to (40) which takesthe integration of the control sequence uv,i and tracking error sequence σi

over the entire control interval [0, Tf ]. In the sequel, the integration processes(40), (41) play the role of an averaging operation on the two noisy sequencesuv,i and σi and are able to remove the majority of those high frequency com-ponents. This averaging operation is especially important to uv,i-updatingwhich usually contains more high frequency components.

Remark 3. In practical implementation, both existing VSC and iterative learn-ing schemes can only effectively control the system in the frequency band[0, ωs/2]. Based on the same reason, in this chapter, the Fourier series ofueq, σi, uv,i, ul,i and νi−1 are truncated into summations from 0 to N in-stead of from 0 to ∞.

7 Illustrative Example

The nonlinear dynamical system described below was chosen for simulationx1 = x2

x2 = f(x1, x2) + b(x1, t)u

where f(x1, x2) = x1 sin(x2) + 10 sin(10πt) and b(x1, t) = 2 + cos(x1). Thedesired trajectory is x1,d = 0.2−0.2 cos(2πt). The switching surface is definedby σ = (x1,d−x1)+(x1,d−x1). The initial values are x1(0) = 0, x1(0) = 0 andσ(0) = 0 is satisfied. The sampling interval is Ts = 1 msec. It is assumed thatfmax = 10+|x1| and bmin = 1 are known a priori. It can be seen that f(x1, x2)and b(x1, t) are Lipschitzian which satisfies A2. According to (8), a typicalVSC is implemented as uv = b−1

min [fmax + |x1,d + x1,d − x1|+ 0.01] sat(σ, 0.2).Fig.3 shows the steady-state tracking error is bounded by 9 × 10−3. To

improve tracking accuracy, the learning algorithms developed in Sections IV,V and VI were individually tested. The learning rate is βl = 30 for learn-ing with σi-updating and N = 20 is chosen for frequency domain learning.For comparison, Fig.4 and Fig.5 plot together the maximum tracking er-ror over iterations of σi− and uv,i−updating for the learning in the time


domain and frequency domain respectively. It is shown that the developedlearning algorithms effectively improve the tracking accuracy. uv,i−updatingachieves faster convergence than σi−updating as uv,i better reflects the de-mand from the control tasks. Frequency domain learning further improvestracking accuracy since the integration process (40) nullifies the majority ofhigh frequency components caused by quantization error and other non-idealfactors due to limited sampling frequency. Learning in the frequency domainwith uv,i-updating obtains the fastest convergence. It is shown in Fig.6 thatthe tracking error has been reduced to below 12 × 10−7 during the 10−thiteration and Fig.7 shows the control profile is fairly smooth.

8 Conclusion

In this chapter, a new control approach - Learning Variable Structure Control(LVSC) is proposed for repeatable tracking control tasks. LVSC is constructedby simply adding an iterative learning mechanism to VSC with a smoothingfunction and the implementation is easy. By rigorous proof we show that theLVSC scheme makes the tracking error converge uniformly to zero, systemstates converge uniformly to the desired trajectories, and bounded learn-ing control converge to the equivalent control almost everywhere. The LVSCscheme also retains the insensitivity property to system uncertainties. Imple-mentation of the proposed learning mechanism by means of Fourier seriesexpansion enhances the robust property of LVSC and improves tracking per-formance. Simulation results show the effectiveness of the proposed LVSCapproaches.

References

1. Utkin, V. I. (1992) Sliding Modes in Control Optimization. Springer-Verlag,Berlin

2. Fu, L. C. and Liao, T. L. (1990) Globally stable robust tracking of nonlinearsystems using variable structure control and with an application to a roboticmanipulator. IEEE Transactions on Control Systems Technology, 35, 1345-1350

3. Young, K. D. (1997) Sliding-mode Design for Robust Linear Optimal Control.Automatica, 33, 1313-1323

4. Yu, X. and Man, Z. (1998) Multi-input Uncertain Linear Systems with TerminalSliding-mode Control. Automatica, 34, 389-392

5. Xu, J. X. and Cao, W. J. (2000) Synthesized sliding mode control of a single-link flexible robot. International Journal of Control, 73, 197-209

6. Slotine, J. J. E. (1984) Sliding Controller Design for Nonlinear Systems. Inter-national Journal of Control, 40, 421-434

7. Yoo, D. S. and Chung, M. J. (1992) A variable structure control with sim-ple adaptation laws for upper bounds on the norm of the uncertainties. IEEETransactions on Control Systems Technology, 37, 860–864

8. Yao, B. and Chan, S. P. and Wang, D. (1994) Variable structure adaptivemotion and force control of robot manipulators. Automatica, 30, 1473-1477

52 J.-X. Xu

9. Xu, J. X. and Cao, W. J. (2000) Synthesized sliding mode and time-delaycontrol for a class of nonlinear uncertain systems. Automatica, 36, 1909-1914

10. Bartolini, G., Ferrara, A. (1999) On the parameter convergence properties ofa combined VS/adaptive control scheme during sliding motion. IEEE Transac-tions on Automatic Control, 44, 789-793

11. She, J.H., Pan, Y.D. and Nakano, M. (2000) Repetitive Control System withVariable Structure Controller. Proceedings of the 6th IEEE International Work-shop on Variable Structure Systems, 273–282

12. Bein, Z. and Xu, J.-X., (1998) Iterative Learning Control - Analysis, Design,Integration and Applications. Kluwer Academic Publishers (edited).

13. (2000) Sepcial Issue on Iterative Learning Control. International Journal ofControl, 73, 819-999.

14. Griffel, D.H. (1981) Applied Functional Analysis (pp. 108-110). John Wiley &Sons, New York

15. Ioannou, P.A. and Sun, J. (1996) Robust Adaptive Control (pp. 71 and pp.101-104). PH, Englewood Cliffs, New Jersey

16. Miller, R.K. and Michel, A.N. (1982) Continuation of solutions. In: OrdinaryDifferential Equations. (pp. 49–68). Academic Press, New York

17. Voxman, W.L. and Goetschel, R.H. (1981) Advanced Calculus, An Introductionto Modern Analysis. (pp. 149–150). Marcel Dekker, Inc, New York

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−4

−2

0

2

4

6

8

10x 10

−3

Time (Second)

Tra

ckin

g E

rror

Fig. 3. x1,d − x1, tracking error for a typical VSC


0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

Iteration Sequence

Max

imum

Tra

ckin

g E

rror

Fig. 4. Maximum tracking error over each iteration for σi−updating, dashed line– time domain learning; solid line – frequency domain learning.

1 2 3 4 5 6 7 8 9 1010

−6

10−5

10−4

10−3

10−2

Iteration Sequence

Max

imum

Tra

ckin

g E

rror

Fig. 5. Maximum tracking error over each iteration for uv,i−updating, dashed line– time domain learning; solid line – frequency domain learning.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−12

−10

−8

−6

−4

−2

0

2

4

6x 10

−7

Time (Second)

Tra

ckin

g E

rror

Fig. 6. x1,d−x1, tracking error during the 10−th iteration under frequency domainlearning with uv,i−updating.

54 J.-X. Xu

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−8

−6

−4

−2

0

2

4

6

Time (Second)

Con

trol

Inpu

t

Fig. 7. Control profile during the 10−th iteration under frequency domain learningwith uv,i−updating.


Discrete-time Variable Structure Control

Katsuhisa Furuta1 and Yaodong Pan2

1 Department of Computers and Systems Engineering, Tokyo Denki University,Hiki-gun, Saitama 350-0394, Japan

2 Department of Electrical Engineering, The Ohio State University,205 Dreese Laboratory, 2015 Neil Avenue, Columbus, OH 43210-1272, USA

Abstract. This chapter presents some discrete-time variable structure (VS) con-trol design algorithms with sliding sectors and then proposes a discrete-time VScontroller with an invariant sliding sector. The invariant sliding sector is an in-variant subset of the state space determined by a linear and a quadratic functionson the state variable. To ensure the invariance, a VS control law is implemented.Inside the sector with the VS control law, a Lyapunov function keeps decreasing. Ifthe state is inside the subset in some time instance, then the VS control input willlet the state remain inside. A discrete-time VS controller with the invariant slidingsector for discrete-time systems is designed such that the state is moved into thesector in finite steps and stays there from then on. The resultant VS control sys-tem is quadratically stable as there exists a Lyapunov function which decreases inthe state space. Simulation results are given to show the efficiency of the proposeddesign algorithm of the discrete-time VS controller.

1 Introduction

The Variable Structure Control (VSC) system has been mainly consideredfor continuous-time systems in the form of sliding mode [25]. When it is im-plemented in practical systems by digital controllers, not only may the chat-tering be generated around the sliding mode because of the finite switchingfrequency, but the stable sliding mode designed for continuous-time systemsmay also become unstable after discretizing [8]. Therefore it is important andalso necessary to investigate the properties of the VS controller with slidingmode after discretization. A number of research papers that appear in recentliterature have been devoted to the implementation of continuous-time VScontrol by computer or via discretization and to the design of discrete-timeVS controllers.

Milosavljevic[16] proposed a concept of quasi-sliding mode and gave anecessary reaching condition

(sk+1 − sk)sk < 0

to ensure the existence of the quasi-sliding mode for discrete-time VS controlsystems, which is similar to the reaching condition

s(x)s(x) < 0


applied to continuous-time VS control systems. Sarpurk et al[21][15] gave asufficient reaching condition

|sk+1| < |sk|and pointed out that the control input must have upper and lower boundsdepending on the distance of the system state from the hyperplane. An equiv-alent reaching condition, i.e.

|sk+1sk| < s2kmay be found in Sira-Ramirez’s paper[22].

Furuta[4] proposed another equivalent reaching condition:

sk(sk+1 − sk) < −12(sk+1 − sk)2

and defined a sliding sector as

|sk| ≤ 12|GΓ |f0

n∑i=1

|xi(k)|

where h is a positive constant and xi(k) (i = 1, 2, · · · , n) is the i-th elementof the state variable xk. The equivalent control in [4] is obtained by setting

sk+1 = sk

while in the β equivalent control approach[6], the equivalent control is foundwith

sk+1 = βsk

where |β| < 1. The state will move into the sliding sector because the pro-posed discrete-time VS control input satisfies the above reaching conditionoutside the sliding sector. Inside the sliding sector, the equivalent control lawensures the stability of the closed-loop system. With the consideration of pa-rameter uncertainties, a robust discrete-time VS controller was also designedin [4] where the sliding sector was modified as

|sk| ≤ 12|GΓ |(f0 + d)

n∑i=1

|xi(k)|

where the positive constant d is the upper bound of the parameter uncer-tainty. Furuta [5] also proposed a discrete-time VS controller with slidingsector for those systems described by a transfer function, where the self-tuning control is used to ensure the robust stability.

Drakunov and Utkin[3] suggested for sampled-data systems that somesliding manifold must be reached in a finite time interval and the systemrepresentative point after reaching the sliding manifold must be confined tothat region to ensure the existence of discrete-time sliding mode. Yu and

58 K. Furuta and Y. Pan

Potts[27] investigated the pseudo-sliding mode behavior of the discrete-time VS control system where a class of n-th order linear systems with a VScontroller was discretized.

Similar to the sliding sector proposed by Furuta [4], Wang et al [26] defineda switching region, which is determined by

−(δ + d) ≤ 2skC2B2

∑ni+1 |xi(k)| ≤ (δ + d)

and designed a discrete-time VS controller based on the region such thatnot only is the discrete-time uncertain system stabilized robustly but thechattering along the sliding mode is also reduced explicitly.

Gao et al [10] defined a quasi-sliding mode band

|s(k)| < εT

1− qTwhere T is the sampling interval and ε and q are positive constants, whichsatisfy

0 < 1− qT < 1.

It is pointed out that the plant states oscillate between the limited band withthe so-called reaching law

s(k + 1) = (1− qT )s(k)− εT sgn(s).which is obtained by approximating the following continuous-time reach law.

s(t) = −εsgn(s(t))− qs(t),Koshkouei and Zinober proposed discrete-time sliding mode controllers

for signal input systems[13] and multi-input systems[14] with a lattice-wise hyperplane, on which there is a countable set of points comprisinga so-called lattice. They separated the sequence s(k)∞k=0 into two subse-quences s+(k)∞k=0 and s−(k)∞k=0, which are the positive and negativesubsequences, respectively. Then sufficient conditions for the existence of thesliding mode were given by

s+(k + 1) < s+(k)s−(k + 1) > s−(k)

Misawa [18] suggested to use a boundary layer and proposed a discrete-time sliding mode control, which ensures the attractiveness and invarianceof the boundary layer. Some discrete-time VS controllers were designed withobserver[17][24] or adaptive control algorithms[1] or uncertain estimation[12].And some others were designed for sampled-data systems by discretizingcontinuous-time VS controllers, for example, [2][23][11]. It is pointed out thatthe use of a discontinuous control law in a sampled-data system will result

59Discrete-Time Variable Structure Control

in chattering phenomenon in a boundary layer with thickness O(T 2) whereT is the sampling interval [23] and the robustness of the VS controller canbe improved by decreasing the sampling interval T [11].

As an alternate design algorithm of the Variable Structure Control (VSC),a PR-sliding sector has been proposed in the design of VSC for a chatteringfree controller and for the implementation in discrete-time control systems[9].The PR-sliding sector in [9] is defined as

|sk| < δk =√xT

kQxk,

which is a subset on the state space where Q is a positive semi-definite matrix.Inside the sector, P -norm of state defined as

||xk||P =√xT

k Pxk

decreases with zero control and satisfies

||xk+1||2P − ||xk||2P < −||xk||2Rwhere P and R are positive definite matrices, which implies why it is calledthe PR-sliding sector. The corresponding VS controller was designed suchthat the state moves from the outside to the inside of the sector with someVS control law, the control input is zero inside the sector, and some Lyapunovfunction keeps decreasing in the state space with specified negativity of itsderivative.

The proposed continuous-time PR-sliding sector theoretically is an in-variant subset. But inside the PR-sliding sector, there exists a potentialpossibility for the state to move out of the sector with zero control input.Therefore the invariance may be lost if the switching frequency is finite al-though it does not affect the stability of the proposed continuous-time VSCsystem. The discrete-time PR-sliding sector is not an invariant subset as theswitching frequency is finite in a discrete-time VS control system.

The invariant sliding sectors for continuous-time[19] and discrete-time[20] systems have been proposed. The invariant sliding sector is a PR-sliding sector and an invariant subset of the state space. Inside the invariantsliding sector, some norm of state decreases because it is a PR-sliding sector.And especially inside the sector, the state will not move out of the subsetwith some control law. The proposed VS controllers with the invariant PR-sliding sector ensures that the state moves from the outside to the inside ofthe sector in a finite time, stays inside it forever after being moved into it,and some Lyapunov function keeps decreasing in the state space with someVS control law. The resulted VS control system thus is quadratically stableand without chattering. Such invariant sliding sector for continuous-time sys-tems remains invariant even if the switching frequency is finite. Therefore itis possible to be implemented in digital systems.


In [28], Yu and Yu also discussed the existence of an invariant slidingsector for a second order discrete-time system and conditions to guaranteethe existence were derived.

In this chapter, definitions and design algorithms of the discrete-time slid-ing sector and PR-sliding sector will be presented at first. Then the discrete-time VS control algorithm using transfer function will be discussed. Finallyan invariant discrete-time sliding sector will be defined. A VS controller withthe proposed invariant sliding sector for discrete-time systems will be pro-posed.

The organization of the paper is as follows. Section 2 presents the discrete-time VS control with sliding sector[4]; Section 3 presents the discrete-timeVS control using transfer function[7]; Section 4 presents the discrete-time VScontrol with PR-sliding sector[9]; Section 5 defines the invariant discrete-timesliding sector, gives the design algorithm and proposes the discrete-time VScontroller with the invariant sliding sector; and Section 6 gives the simulationresult.

2 VS Control with Sliding Sector

A single-input discrete-time plant described by the following state equationis considered in this chapter except Section 3 where transfer function is used.

xk+1 = Φxk + Γuk (1)

where xk ∈ Rn and uk ∈ R1 are state and input vectors, respectively; Φ andΓ are constant matrices of appropriate dimensions; and the pair (Φ, Γ ) iscontrollable.

Similar to the continuous-time sliding mode control, a function sk is de-fined as

sk = Gxk. (2)

where G is designed so that the state staying on sk = 0 for all k is stable asshown in the following lemma and theorem.

Lemma 1. [4] The equivalent control law for the system (1) such that thestate rests on the hyperplane of sk = sk+1 for all k is given by

uk = Feqxk (3)

where

Feq = −(GΓ )−1G(Φ− I). (4)

Theorem 1. [4] The hyperplane, equivalently G, should be determined sothat the system

xk+1 = [Φ− Γ (GΓ )−1G(Φ− I)]xk,

Gxk = 0,

is stable.


As the second step, the control law to transfer the state on the hyperplaneshould be determined. Let Vk and ∆sk be respectively defined as

Vk =12s2k

∆sk+1 = sk+1 − skThen the control law to decrease Vk is given by the following lemma.

Lemma 2. [4] If the control satisfies

sk∆sk+1 < −12(∆sk+1)2, for sk = 0 (5)

then

Vk+1 < Vk

In this section, the following type of control law is considered:

uk = (Feq + FD)xk, (6)

where Feq is given by (4) and FD is a discontinuous control law. Then thevariable structure control law to make the system stable is given by thefollowing theorem.

Theorem 2. [4] If the sliding mode satisfies the conditions of Theorem 1,and the control is chosen by (6), where the absolute value of the i-th elementof FD, fi is constant for all time and the same for all i, i.e.,

|f+i | = |f−i | = f0, i = 1, 2, · · · , n,

then the control law

fi =

f0 for (GΓ )skxki < −δi,0 for −δi ≤ (GΓ )skxki ≤ δi,

−f0 for (GΓ )skxki > δi,(7)

makes the system stable, where xki is the i-th element of xk and δi is definedas

δi =12(GΓ )2

n∑j=1

|xki||xkj |f0

with the amplitude of f0 limited by

0 < f0 < | 2GΓ

∑nj=1 |t1j | |,

t1i being the i-th element of t1 satisfying Gt1 = 1, Gti = 0, ti ⊥ tj (i = j).


With the consideration of the parameter uncertainty, represent the actualsystem matrix Φ as

Φ = Φ0 +∆Φ

where Φ0 is known and ∆Φ is the uncertainty of the system matrix, which isrepresented by

∆Φ = ΓD, (8)D =

[d1 d2 · · · dn

], |di| < d(i = 1, 2, · · · , n). (9)

The known system is represented by

xk+1 = Φ0xk + Γuk. (10)

It is assumed that both (1) and (10) are stabilized on sk = 0. Then arobust discrete-time VS controller is designed by the following theorem.

Theorem 3. [4] If the plant system matrix Φ has the uncertainty ∆Φ sat-isfying (8) and (9), and on sk = 0, the plant is stable with the equivalentcontrol, then the i-th element fi of the control law FD satisfying

fi =

f0 for (GΓ )skxki < −δ′i,0 for −δ′i ≤ (GΓ )skxki ≤ δ′i,

−f0 for (GΓ )skxki > δ′i,

(11)

makes the system stable, where

δ′i =12(GΓ )2

n∑j=1

|xki||xkj |(f0 + d)

with the amplitude of f0 limited by

d < f0 < | 2GΓ

∑nj=1 |t1j | | − d.

3 VSC for Discrete-time Input-Output System

This section considers a single input and single output system. The followingdiscrete relation represents the controlled plant with input uk, disturbancewk and output yk,

A(q−1)yk = q−dB(q−1)uk +D(q−1)wk, (12)

where A(q−1) and B(q−1) have no common factors, q denotes the time shiftoperator defined by

q−tyk = yk−t


and q−d is the pure time delay of the system, d(d ≥ 1) is an integer. A(q−1)and B(q−1) are assumed known and representing

A(q−1) = 1 + a1q−1 + a2q−2 + · · ·+ anq−n,

B(q−1) = b0 + b1q−1 + b2q−2 + · · ·+ bmq−m (b0 = 0),

The objective of the control is that the output yk tracks the reference rk inthe presence of the disturbance. The polynomial models of the reference andthe disturbance are assumed to be

Ψr(q−1)rk = 0, Ψw(q−1)wk = 0.

Let Ψ(q−1) be the least common multiple of Ψr(q−1) and Ψw(q−1) then

Ψ(q−1)rk = 0, Ψ(q−1)wk = 0,

where Ψ(q−1) is coprime with A(q−1), B(q−1). Subtract A(q−1)rk from (12)and multiplying Ψ(q−1) to both sides, the following relation can be obtained

A(q−1)Ψ(q−1)ek = q−dB(q−1)Ψ(q−1)uk, (13)

where the error will be defined as

ek = yk − rk.

3.1 Servo Control of Generalized Minimum Variance Control

The objective of the control in this subsection is to minimize the generalizedvariance of the controlled variables sk+d, that is, in the deterministic case, togive the control input satisfying

sk+d = C(q−1)ek+d +Q(q−1)Ψ(q−1)uk = 0, (14)

where real polynomials C(q−1) and Q(q−1)

C(q−1) = 1 + c1q−1 + c2q−2 + · · ·+ cnq−n,

Q(q−1) = Q0(1 + q1q−1 + q2q−2 + · · ·+ qm−1q−m+1),

are determined so that the error vanishes if the above (14) is satisfied. Atfirst, the generalized minimum variance control without using VSS will bediscussed. The equation (14) is rewritten as

sk+d = (E(q−1)B(q−1) +Q(q−1))Ψ(q−1)uk + F (q−1)ek= G(q−1)Ψ(q−1)uk + F (q−1)ek, (15)

where E(q−1) and F (q−1) are polynomials determined to satisfy

C(q−1) = A(q−1)Ψ(q−1)E(q−1) + q−dF (q−1) (16)

and G(q−1) is defined as

G(q−1) = E(q−1)B(q−1) +Q(q−1). (17)


The control input to make sk+d = 0 is, therefore, given by

uk = −[G(q−1)Ψ(q−1)]−1[F (q−1)ek]. (18)

This subsection considers the use of variable structure control in additionto the above conventional generalized minimum variance control so that theoutput satisfies sk = 0. C(q−1) and Q(q−1) should be chosen to satisfy thefollowing lemma.

Lemma 3. [5] The necessary and sufficient condition that the output withzero reference making sk+d = 0 stable is that all zeros of

A(q−1)Q(q−1)Ψ(q−1) +B(q−1)C(q−1) = 0 (19)

are inside the unit disk.

Instead of (18), the following input is considered to be used,

uk = −[G(q−1)Ψ(q−1)]−1[F (q−1)ek − βsk − vk], (20)

where

0 < β ≤ 1.

Substituting (20) into (12) yields

sk+d = vk + βsk. (21)

The auxiliary control input vk is chosen as the state feedback with the variablecoefficients.

vk = h0ek + h1ek−1 + · · ·+ hn−1ek−n+1. (22)

The control input with vk = 0 is called the β equivalent control where sk+d =βsk. The control law given as follows gives a stable system.

Theorem 4. [5] For the plant (12), if the coefficients of the feedback controllaw are chosen

hi =

h skek−i < −δi0 | skek−i | ≤ δi (i = 0, 1, · · · , n− 1),−h skek−i > δi

(23)

then the control system becomes stable, where

δi = ηn−1∑j=0

| ek−i || ek−j | h (24)

and

η ≥ α

2(αβ − α+ 1)(25)

and it is assumed that α ≥ 1.


3.2 Self-Tuning Servo Control based on VSS

Now it is assumed that the plant (12) has parameter uncertainty. The givenplant is considered to have the known delay d in the plant. Parameters ai, biof A(q−1) and B(q−1) are, however, assumed unknown except b0 (= 0). Inthis subsection, the control algorithm is determined based on generalizedminimum variance control. Different control laws and parameter identificationmethods are employed inside and outside the sector. The sector is defined as

Sk = sk| | sk |≤((1− β)p +

√(1− β)2p2 + 2p2γ

γ

)(|φk|), (26)

where

γ =2α(1− β)− (1− β)2,

|φk| =n−1∑j=0

|ek−j |+m+d−1∑

j=1

|Ψ(q−1)uk−j |

and p will be defined later in (34). In the outside of the sector, control and pa-rameter identification are done simultaneously based on the Lyapunov func-tion. The polynomial sk is defined by (14). When parameters A(q−1), B(q−1)are known, sk is given by (15). When A(q−1) and B(q−1) are unknown,G(q−1) and F (q−1) cannot be obtained exactly. In this case, the control in-put uk is determined by using the estimate of G(q−1) and F (q−1), denotedby G(q−1) and F (q−1), as follows. For the outside of the sector,

uk = −Gk(q−1)Ψ(q−1)−1[Fk(q−1)ek − βsk

−n−1∑j=0

hjek−j −m+d−1∑

j=1

wjΨ(q−1)uk−j

, (27)

where hj, and wj are nonlinear bang-bang type functions depending onthe state outside the sector and take values out of h, 0, − h. Since b0 isassumed known, g0 can be given. Let the estimate of G(q−1) and F (q−1) be

θk = θk−d + Γ−1φksk (Γ > 0), (28)

θ, φk are defined as

θ = [f0, f1, · · · , fn−1, g1, · · · , gm+d−1]T

φk =[ek, ek−1, · · · , Ψ(q−1)uk−1, · · · , Ψ(q−1)uk−m−d+1

]T

For the inside of the sector;

uk = −Gk(q−1)Ψ(q−1)−1[Fk(q−1)ek

]. (29)

The following main theorem establishes the stability of the closed loop sys-tem.


Theorem 5. [5] The control system, which is employed for the outside ofthe sector Sk where the control and the simultaneous parameter estimationare determined by (27) and(28) respectively, brings either the system into thesector or the error to zero. The coefficients of (27), hi and wi are givenby the following relations:

hi =

h skek−i < −δi0 |skek−i| ≤ δi−h skek−i > δi

(30)

(i = 0, 1, · · · , n− 1)

wi =

h skΨ(q−1)uk−i < −σi

0 |skΨ(q−1)uk−i| ≤ σi

−h skΨ(q−1)uk−i > σi

(31)

(i = 1, 2, · · · ,m+ d− 1)

where

δi = ηa|ek−i|n−1∑

j=0

|ek−j |+m+d−1∑

j=1

|Ψ(q−1)uk−j |+n−1∑j=0

|rk−j |h

(i = 0, 1, · · · , n− 1), (32)σi = ηa|Ψ(q−1)uk−i|

n−1∑j=0

|ek−j |+m+d−1∑

j=1

|Ψ(q−1)uk−j |+n−1∑j=0

|rk−j |h

(i = 0, 1, · · · ,m+ d− 1), (33)

where

ηa ≥ α

1− α+ αβ,

p is the upper bound of the uncertainty of the parameters defined as

maxi|θi − θki| < p, (34)

where θki denotes the i-th element of θk and α > 1.The following Lyapunovfunction

Vk =12s2k +

12θTk−dΓ θk−d (35)

decreases outside the sector. By using the control (29), inside the sector, thestability of the closed loop system is assured.


4 VS Control with PR - Sliding Sector

4.1 PR - Sliding Sector

Consider the discrete-time system described by the state equation (1).

Definition 1. [9] The P -Norm || · ||P of the discrete-time system state isdefined as

||xk||P = (xTk Pxk)

12 , xk ∈ Rn (36)

where P is a n× n positive definite symmetric matrix.

The square of the P -norm is denoted as

Lk = ||xk||2P = xTk Pxk > 0, ∀xk ∈ Rn, xk = 0 (37)

where the positive definite matrix P will be chosen later by Theorem 1.If the autonomous system given by equation (1) is quadratically stable,

then there exists a positive definite symmetric matrix P and a positive semi-definite symmetric matrix R = CTC such that

∆Lk = Lk+1 − Lk = xTk (Φ

TPΦ− P )xk ≤ −xTkRxk, ∀xk ∈ Rn

where P ∈ Rn×n, R ∈ Rn×n, C ∈ Rl×n, l ≥ 1, and (C,Φ) is an observablepair. For an unstable system, this inequality does not hold. It is possible todecompose the state space into two parts such that one part satisfies thecondition

∆Lk > −xTkRxk

for some element xk ∈ Rn, and the other part satisfies the condition

∆Lk ≤ −xTkRxk

for some other element xk ∈ Rn. The latter elements form a special subset inwhich the P -norm ||xk||P decreases with zero input. Accordingly, this specialsubset is defined as a PR-sliding sector.

Definition 2. [9] A Discrete-time PR-Sliding Sector is defined on the statespace Rn as

S = xk| |sk| ≤ δk, xk ∈ Rn (38)

inside which the P -norm decreases with zero input and the difference of thecandidate Lyapunov function Lk (37) satisfies the condition:

∆Lk = Lk+1 − Lk = s2k − δ2k − xTkRxk ≤ −xT

kRxk, ∀xk ∈ S.where P is a n× n positive definite symmetric matrix, R is a n× n positivesemi-definite symmetric matrix, R = CTC, C ∈ Rl×n, l ≥ 1, and (C,Φ) isan observable pair. The linear functional sk and the square root δk of thequadratic function δ2k are respectively determined in the form as follows:

sk = Sxk, S ∈ R1×n, (39)

δk =√xT

kDxk,D(≥ 0) ∈ Rn×n (D = 0). (40)


Such sliding sector is a subset around a hyperplane sk = 0 and betweentwo surfaces determined by |sk| = ±δk. The way to choose S and D usingRiccati Equation

P = Q+ ΦTPΦ− ΦTPΓ (1 + ΓTPΓ )−1ΓTPΦ (41)

will be presented as follows:

Theorem 6. [9] If the positive definite symmetric solution P of the discrete-time Riccati equation (41) is used to define the P -norm and the positive semi-definite symmetric matrix R ∈ Rn×n is chosen so that D = Q − R ≥ 0 andD = 0, then the parameters of the PR-sliding sector (38) are determined by

sk = Sxk, S = ΓTPΦ/√1 + ΓTPΓ

δk =√xT

kDxk, D = Q−Rwhere Q is the positive definite symmetric matrix in the discrete-time Riccatiequation (41).

Remark 1. For simplicity, the parameter matrices Q, P , R, and D of thePR-sliding sector (38) may be chosen using the following steps:

1. Choose a n× n positive definite symmetric matrix as Q,2. Solve the discrete-time Riccati equation (41) for the positive definite sym-

metric solution P ,3. Choose a positive constant r ( 0 < r < 1) and let D = rQ and R =

(1− r)Q.

4.2 Lazy Control with PR - Sliding Sector

Based on the PR-sliding sector (38), a discrete-time VS controller can bedesigned by the following theorem.

Theorem 7. [9] For any controllable discrete-time plant described by (1),the discrete-time VS control law

uk =0 xk ∈ S,−b−1(Fxk +Ksgn(bsk)δk) xk∈S (42)

enables the system to be quadratically stable if b is invertible, and

0 < K ≤ min1,√1 + hh

|b|K2R > FTF

where the PR-sliding sector S is defined in (38) by using the discrete-timeRiccati equation with D = rQ and R = (1 − r)Q for some positive constantr (0 < r < 1), and

b = SΓ, F = SΦ, h = ΓTPΓ.


With the VS control law (42), it can be ensured that

1. the system state is moved from the outside to the inside of the PR-slidingsector, and

2. the following Lyapunov function

Vk = xTk Pxk + s2k (43)

keeps decreasing in the state space, where P ∈ Rn×n is the solution ofthe discrete-time Riccati equation (41), and sk is the linear functionalused in the definition of the PR - sliding sector S (38).

Therefore the resultant discrete-time VS control system with the PR-slidingsector is quadratically stable. And especially as the control input inside thePR-sliding sector is zero, i.e., no control effect is needed inside the sector, wenamed this kind of discrete-time VS control algorithm Lazy Control.

5 VS Control with Invariant P R - Sliding Sector

5.1 Invariant P R - Sliding Sector

The PR-sliding sector presented in the last section is a subset of the statespace Rn around a hyperplane sk = 0 and is bounded by two surfaces sk =±δk. Inside it P -norm decreases with zero control input to ensure the stabilityof the VS control system. Therefore, the followings hold inside the PR-slidingsector with zero control input:

|sk| ≤ δk∆Lk ≤ −xT

kRxk

which determine the form of the sector and the convergence inside the sector,respectively.

An invariant P R-sliding sector is defined to be a PR-sliding sector at firstand then to be invariant. To ensure the invariance, some control law insidethe sector is necessary. Thus the invariant P R-sliding sector

1. has the same form as the PR-sliding sector,2. is a convergent subset, i.e. inside it the P -norm of the state decreases

with some VS control law, and3. is an invariant subset of the state space, i.e., if the state is inside or moves

into the sector in some time instant, then the state will stay inside thesector since then.

Definition 3. An Invariant P R-Sliding Sector S for the discrete-time system(1) with some control input is defined as

S = xk| s2k ≤ δ2k,∆s2k ≤ ∆δ2k,∆Lk ≤ −xTk Rxk xk ∈ Rn (44)


where the discrete-time Lyapunov function candidate Lk is defined as

Lk = xTk P xk > 0, ∀xk ∈ Rnand xk = 0, (45)

sk is a linear function on xk and δk is the square root of a quadratic functionon xk as:

sk = Sxk, S ∈ R1×n

δk =√xT

k Dxk,

P , R, and D are some n× n positive definite symmetric matrices.

Inside the discrete-time P R-invariant sliding sector defined in (44), thefollowing holds:

s2k ≤ δ2k, ∀xk ∈ S (46)s2k+1 ≤ δ2k+1, ∀xk ∈ S (47)

∆Lk ≤ −xTk Rxk, ∀xk ∈ S (48)

where the first inequality determines the form of the sector, the second oneensures the invariance of the sector, and the third one guarantees the stabilityof the sector, i.e. the state inside the sector will converge to the origin.

As the forms of the PR-sliding sector and the invariant P R-sliding sectorare the same, it is possible to design the invariant P R-sliding sector in thefollowing steps:

1. Design a PR-sliding sector using the discrete-time Riccati equation aspresented in the last section and let the hyperplane sk = 0 of the invariantP R-sliding sector be the one of the PR-sliding sector;

2. Design a VS control law such that both the absolute value of the gen-eralized function sk and the P -norm decrease for some positive definitematrices P and R.

3. Modify the sector such that the conditions in (46), (47), and (48) aresatisfied. Thus as the result, an invariant P R-sliding sector is designed.

Based on the PR-sliding sector, to ensure the decreasing of the function|sk|, the following VS control law is often used:

uk = −(SΓ )−1(SΦxk − βsk) (49)

with which s2k decreases as

s2k+1 = β2s2k < s2k

and the closed-loop system of the plant (1) with the VS control law (49) isgiven by:

xk+1 = Φxk (50)


where

Φ = Φ− (SΓ )−1ΓS(Φ− βIn)β (|β| < 1) is a constant and In is the n × n identity matrix. It is obviousthat one of the eigenvalues of the above closed-loop system is equal to theconstant β and others are equal to the eigenvalues of the plant (1) in thehyperplane sk = 0, i.e.

xk+1 = Φxk + Γuk

sk = Sxk = 0 (51)

Therefore if the eigenvalues of the plant (1) in the hyperplane sk = 0 are allinside the unity circle, then the eigenvalues of the closed-loop system (50)are all inside the unity circle. Thus it is possible to find two n × n positivedefinite matrices P and R such that

ΦT P Φ− P = −R, (52)

i.e., with the VS control law (49) the P -norm or the discrete-time Lyapunovfunction decreases as

∆Lk = xTk Rxk ≤ 0.

Moreover if the constant β is chosen to satisfy:

|β| ≤ |λi(Φ)|, (i = 1, 2, · · · , n)where λi(Φ) (i = 1, 2, · · · , n) are the eigenvalues of Φ, then the followinginequality can be easily shown to be true for some positive definite matricesP and R satisfying (52):

(1− β2)P − R ≥ 0 (53)

i.e.

ΦP Φ− β2P ≥ 0

According to the definitions of the PR- and the invariant P R-slidingsectors, the hyperplanes inside the sectors are the same but the boundariesof the sectors, i.e. the parameter matrices D and D of the quadratic functionsδ2k’s are different. Let the invariant P R-sliding sector be a subset of the PR-sliding sector by defining the quadratic function δ2k as

D = γP ≤ D (54)

where D is the parameter used to determine the PR-sliding sector (38) andγ is a positive constant which enables the above inequality hold. Then insidethe invariant sector (44) the condition in (47) holds as:

s2k+1 − δ2k+1 = (s2k+1 − δ2k)−∆δ2k= β2s2k − δ2k + γxT

k Rxk

= β2(s2k − δ2k)− γxTk ((1− β2)P − R)xk

≤ β2(s2k − δ2k) ≤ 0 (55)


The above discussion results in the following theorem.

Theorem 8. A subset of the discrete-time PR-sliding sector in (38) designedby the discrete-time Riccati equation is a discrete-time PR-invariant slidingsector with a VS control law:

uk = −(SΓ )−1(SΦxk − βsk) (56)

if all of eigenvalues of the following matrix

Φ = Φ− (SΓ )−1ΓS(Φ− βIn)are inside the unit circle and the parameters of the sector are chosen as

D = γP < D

where γ is a positive constant, β (|β| < 1) is a constant satisfying

(1− β2)P − R ≥ 0

and P and R are positive definite matrices satisfying

ΦT P Φ− P = R

In fact, an invariant P R-sliding sector can also be designed directly. Asthe conclusion of this section, the steps to design an invariant discrete-timeP R are given as follows:

1. Choose a 1× n matrix S such that the reduced-order systemxk+1 = Φxk + Γuk

sk = Sxk = 0

in the hyperplane sk = 0 is stable;2. Find positive definite matrices R and R and a constant β (|β| < 1)

satisfying:

ΦT P Φ− P = R(1− β2)P − R ≥ 0

where Φ is determined by

Φ = Φ− (SΓ )−1ΓS(Φ− βIn).3. Define the parameter D as

D = γP

where γ is a positive constant, which should be determined with theconsideration of the parameter uncertainty and external disturbance.

Then the sector designed by the above steps satisfies the conditions (46),(47), and (48) of the definition of the invariant P R-sliding sector.


5.2 Variable Structure Controller

With the invariant P R-sliding sector proposed in the last section, a VS con-troller should be designed such that the state will converge to the inside ofthe sector. In this chapter, we propose a VS control law as follows:

uk = −(SΓ )−1(SΦxk − αsk) (57)

where α is a constant which should be chosen to satisfy:

|α| < |β|Considering the VS control law used inside the sector, the VS control lawwith the invariant P R-sliding sector is given as:−(SΓ )−1(SΦxk − βsk), xk ∈ S

−(SΓ )−1(SΦxk − αsk), xk∈S. (58)

Theorem 9. With the above VS control law the state will converge to theinside of the invariant P R-sliding sector in finite steps if the absolute valueof the parameter α is chosen to be small enough. And the resultant VS controlsystem is quadratically stable.

Proof. Outside the invariant P R-sliding sector S, the VS control input isgiven by (57), i.e.

uk = −(SΓ )−1(SΦxk − αsk).Thus the following holds:

s2k+1 − δ2k+1 = (s2k+1 − δ2k)−∆δ2k= α2s2k − δ2k + γxT

k Rxk

= α2s2k − γxTk (P − R)xk

≤ α2s2k − β2δ2k. (59)

It is assumed that the state is outside the sector in time instant k0, i.e.

s2k0> δ2k0

if we choose:

|α| ≤ |β| |δk0 ||sk0 |

then the state will converge into the sector by one step, i.e.

s2k0+1 − δ2k0+1 ≤ α2s2k0− β2δ2k0

≤ 0

After converging into the sector, it has been shown in the last sectionthat the state will remain inside the sector as the sector is designed to be aninvariant one with the Lyapunov function Lk decreasing as:

∆Lk = xTk Rxk ≤ 0.

Therefore the resultant VS control system is quadratically stable.


6 Simulation

Consider a second order continuous-time plant.

x(t) =[0 10 0

]x(t) +

[01

]u(t). (60)

Discretizing it with the sampling interval τ = 0.01 second gives the sampled-data state equation as:

xk+1 =[1 0.010 1

]xk +

[0.000050.01

]uk. (61)

Choose the parameter matrix S of the hyperplane be

S =[1 0.9

]which determines the eigenvalue λ of the discrete-time system (61) in thehyperplane sk = 0 as:

λ = 0.98895

Choose the positive definite matrices P and R as

P =[4.35017939 1.666714251.66671425 1.28644034

]

R =[0.35722967 0.272091790.27209179 0.24018819

]

which satisfy the equation (52). Choose the positive constant γ as

γ = 0.1

then the designed invariant P R-sliding sector for the discrete-time system(61) with the above parameters is shown in Figure 1.

Other parameters of the VS controller used in the simulation are chosenas

α = 0.5β = 0.9

The simulation results are given in Figure 2, 3, and 4, which show thatthe state converges into the invariant P R-sliding sector in a finite time andconverges to the origin inside the sector.

Figure 5, 6, and 7 are the simulation results for the plant with a parameteruncertainty of

A =[0 12 1

], B =

[01.1

]

which show the robustness of the proposed VS controller.


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x2

Hyperplane s(x) = 0

Invariable Sliding Sector

Fig. 1. Invariant P R-Sliding Sector

0 1 2 3 4 5 6 7 8 9 10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1(t) and x2(t)

Time(sec)

x1(t)

x2(t)

Fig. 2. Evolution of State Variables x1(t) and x2(t)


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x2

Hyperplane s(x) = 0

Evolution of State with VSC


Fig. 3. Phase Plane Diagram of x2(t) versus x1(t)

0 1 2 3 4 5 6 7 8 9 10-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

u(t)

Time(sec)

Fig. 4. Evolution of Input u(t)


0 1 2 3 4 5 6 7 8 9 10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x1(t) and x2(t)

Time(sec)

x1(t)

x2(t)

Fig. 5. Evolution of State Variables x1(t) and x2(t) with Parameter Uncertainty

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x2

Hyperplane s(x) = 0

Evolution of State with VSC


Fig. 6. Phase Plane Diagram of x2(t) versus x1(t) with Parameter Uncertainty


0 1 2 3 4 5 6 7 8 9 10-40

-35

-30

-25

-20

-15

-10

-5

0

5

10u(t)

Time(sec)

Fig. 7. Evolution of Input u(t) with Parameter Uncertainty

7 Conclusion

In this chapter, at first we presented some discrete-time VS control designalgorithms using sliding sectors or PR-sliding sector for systems describedby state space equation [4][9] or transfer function[7]. Then we proposed aninvariant P R-sliding sector and the corresponding VS controller for discrete-time systems, with which the state converges into the sector in a finite timeand stays inside it from then on as the designed sector is an invariant subsetof the state space Rn. The resultant VS control system is quadratically stableand chattering-free.

The simulation result shows that the proposed sector is really an invari-ant one and the corresponding discrete-time VS controller has good controlperformance even if there exists parameter uncertainty.

References

1. Bartolini G., Ferrara A., and Utkin V. (1992) Design of discrete-time adaptivesliding mode control. Proc. of the 31st CDC, 2387–2391, Tucson, Arizona

2. Bartolini G., Posano A., and Usai E. (1998) Digital second order sliding modecontrol of SISO uncertain nonlinear systems. Proc. of the American ControlConference, 119–124, Philadelphia


3. Drakunov S. and Utkin V. (1989) On discrete-time sliding mode. IFAC Sym-posium on Nonlinear Control System Design, 484–489

4. Furuta K. (1990) Sliding mode control of a discrete system. System & ControlLetters, 14, 145–152

5. Furuta K. (1993) Vss type self-tuning control. IEEE Trans. on IndustrialElectronics 40, 37–44

6. Furuta K. (1993) VSS-type self-tuning control -β equivalent control approach-.Proc. of the American Control Conference, 980–984, San Francisco

7. Furuta K. and Pan Y. (1993) Discrete-time adaptive vss control system usingtransfer function. Proc. of the 31st Conference on Decision and Control, 1434–1439, San Antonio

8. Furuta K. and Pan Y. (1994) Discrete-time VSS control for continuous-timesystems. Proc. of the First Asian Control Conference, 377–380, Tokyo

9. Furuta K. and Pan Y. (2000) Sliding sectors for VS controller. Automatica,36, 211–228

10. Gao W., Wang Y., and Homaifa A. (1995) Discrete-time variable structurecontrol. IEEE Trans. on Industrial Electronics, 42, 117–122

11. Golo G. and Milosavljevic C. (2000) Robust discrete-time chattering free slidingmode control. System & Control Letters, 41, 19–28

12. Haskara I., Ozguner U., and Utkin V. (1997) Variable structure control foruncertain sampled-data systems. Proc. of the 36th Conference on Decision andControl, 3226–3231, San Diego

13. Koshkouei A. J. and Zinober A. S. I. (1996) Discrete-time sliding mode controldesign. IFAC’96 World Congress, volume G, 481–486, San Francisco

14. Koshkouei A. J. and Zinober A. S. I. (2000) Sliding mode control of discrete-time systems. ASME Journal of Dynamic Systems, Measurement, and Control,122, 793–7802

15. Kotta U. (1989) On the stability of discrete-time sliding mode control system.IEEE Trans. on Automatic control, 34, 1021–1022

16. Milosavljevic C. (1985) General conditions for the existence of a quasislid-ing mode on the switching hyperplane in discrete variable structure systems.Automation Remote control, 46, 307–314

17. Misawa E. (1995) Observer-based discrete-time sliding mode control with com-putational time delay: The linear case. Proc. of the American Control Confer-ence, 1323–1327, Seattle, Washington

18. Misawa E. (1997) Discrete-time sliding mode control: The linear case. ASMEJournal of Dynamic Systems, Measurement, and Control, 119, 819–821

19. Pan Y., Furuta K., and Hatakeyama S. (1999) Invariant sliding sector forvariable structure control. Proc. of the 38th IEEE Conference on Decision andControl, 5152–5157, Phoenix

20. Pan Y., Furuta K., and Hatakeyama S. (2000) Invariant sliding sector fordiscrete-time variable structure control. Proc. of the 3rd Asian Control Con-ference, Shanghai

21. Sarpurk S., Istefanopulos Y., and Kaynak O. (1987) On the stability of discrete-time sliding mode control system. IEEE Trans. on Automatic Control, 32,930–932

22. Sira-Ramirez H. (1991) Nonlinear discrete variable structure systems in quasi-sliding mode. Int. J. Control, 54, 1171–1187


23. Su W., Drakunov S. V., and Ozguner U. (2000) An o(t2) boundary layer insliding mode for sampled-data systems. IEEE Trans. on Automatic Control,45, 482–485

24. Suzuki T., Pan Y., and Furuta K. (1999) Discrete-time VS controller designbased on input-output model. Proc. of the 38th IEEE Conference on Decisionand Control, 3906–3911, Phoenix

25. Utkin U. (1992) Sliding Modes in Control and Optimization. Springer-Verlag26. Wang W., Wu G., and Yang D. (1994) Variable structure control design for

uncertain discrete-time systems. IEEE Trans. on Automatic Control, 39, 99–102

27. Yu X. and Potts P. (1991) A class of discrete variable structure systems. Proc.of 30th CDC, 1367–1372, Brighton, UK

28. Yu X. and Yu S. (2000) Discrete sliding mode control design with invariantsliding sectors. ASME Journal of Dynamic Systems, Measurement, and Control,122, 776–782


Higher-Order Sliding Modes for theOutput-Feedback Control of NonlinearUncertain Systems

Giorgio Bartolini1, Arie Levant2, Alessandro Pisano1, and Elio Usai1

1 Universita degli Studi di CagliariDipartimento di Ingegneria Elettrica ed Elettronica, Piazza D’ArmiI-09123 Cagliari, Italy

2 Institute for Industrial Mathematics4/24 Yehuda Ha-Nachtom St.Beer-Sheva 84311, Israel

Abstract. This chapter examines some aspects of the output-feedback controlproblem for nonlinear uncertain plants, with special emphasis on possible applica-tions of recent results about higher order sliding modes (HOSMs). This regime isestablished when the simultaneous, finite-time, zeroing of an output quantity (thesliding quantity), and of a certain number of its derivatives, is ensured. In thiswork, for any step of an output feedback variable structure control design, namely,the definition of the sliding variable, the synthesis of the control law, and the stateestimation, a survey of proposals characterized by a finite-time convergence tran-sient is presented. Some different types of sliding surfaces in the state space, suchthat the associated constrained motion is characterized by a finite-time convergingdynamics, are recalled. The use of a discontinuous control to make them attractiveand invariant is then analyzed. Finally, real-time differentiators based on HOSMsfor estimating the output derivatives are considered. The twofold objective of thepresent chapter is to survey the most recent results on HOSMs and to highlight theirpossible role in improving existing approaches, to motivate and to draw possiblelines for future research.

1 Introduction

In recent years various systematic procedures and methods to design output-feedback nonlinear control systems have been presented in the literature[2,13,14,20,21,38-46].

In real life, an obvious task is to cope with uncertainties, and an impor-tant contribution to this topic was given by the combined introduction ofsaturated inputs and high-gain observers (HGOs), for estimating the outputderivatives, performed by Khalil and co-workers [2,21], improving previousresults presented in Tornambe [45,46]. The very idea of combining saturat-ing control and high-gain observers proved to be essential for a number ofsuccessive developments.

By using, as feedback signals, the output and a certain number of itsderivatives, possibly estimated, the evaluation of the diffeomorfic transfor-


mation required to put the system in normal form [19] could be avoided. Inparticular, all the uncertainties appearing in the system equation are reducedto satisfy the matching condition [19,21,47].

Outstanding results regarding the semiglobal stabilization of nonlinearuncertain systems using only output measurements were presented by Teeland Praly in [43,44], exploiting a complete uniform observability property.They presented two general backstepping lemmas that can be applied torather general classes of systems in order to achieve semiglobal stabilization.

In a recent work by Atassi and Khalil [2], under the hypothesis thata stabilizing globally bounded (possibly dynamic) state-feedback control isavailable, it was demonstrated that using a high gain observer one can re-cover the performance achieved under state feedback; thus a rather generalnonlinear separation principle was established.

Few results are available regarding the output-feedback control of non-minimum phase systems [46], which seems to be one of the most challeng-ing problems arising from the status of the art of nonlinear control theory.Recently, Isidori presented a quite general framework for nonlinear robustoutput-feedback control design [20], and his result encompasses systems withunstable zero-dynamics.

We focus the present chapter on the finite-time output-feedback controlof systems having globally-defined relative degree and stable zero dynamics;in particular, a way to output feedback provided by the sliding-mode controlapproach is described [47,40,35,8].

Our aim is to contribute to put the basis for possible enhancements of therecently developed theory, in order to cope with problems that are still par-tially unsolved: the presence of non-parameteric uncertainties and/or unmod-eled dynamic actuators, the robustness against measurement noise (dramaticwhen multiple output differentiation is required), the transient peaking, thecounteraction of finite escape-time, etc.

In this context we highlight the role of higher-order sliding modes (HOSMs)as a possible alternative, or a complement, to existing methods, in orderto deal with the challenging problem of designing output-feedback controlschemes for nonlinear systems of high relative-degree affected by heavy modeluncertainties.

The chapter is organized as follows: in next Section 2 a formal statementof the problem, and the main standing assumptions, are given. Then, inSection 3, some simulation examples, which identify the presence of a “cost”associated with nonlinear output-feedback schemes analogous to that of thelinear case [18], are discussed. In Section 4 the basic concepts and definitionsregarding higher order sliding modes are recalled for the readers’ convenience,while in Section 5-7 recent results regarding sliding constraint, controller, andstate-observer design are discussed, respectively. Simulations are reportedthroughout the paper.

84 G. Bartolini et al.

2 Problem Statement

Consider the nonlinear SISO system

x = f(x) + g(x)uy = h(x) (1)

with unavailable state vector x ∈ Rn, control variable u ∈ R and measurableoutput y ∈ R. Let f , g and h be unknown smooth vector-fields of properdimension.

Systematic approaches for nonlinear output feedback control design inpresence of various type of uncertainties can be found in [19,47,21,23,37].

Actually, the heavy uncertainty of the problem prevents immediate reduc-tion of (1) to any standard form by means of standard approaches based onthe knowledge of f , g and h. The drift and control vector fields f and g, andthe output map h, are unknown, but they satisfy proper growth conditionsto be specified.

If system (1) possesses a globally-defined relative degree r [19], then theinput-output dynamics turn out to be expressed as

y(r) = Lrfh(x) + LgL

r−1f h(x)u (2)

where Lg, Lf are Lie derivatives, and condition LgLr−1f h(x) = 0 holds, glob-

ally, by assumption (see [19]). Put ξ = [y, y, . . . , y(r−1)].It is always possible to define a set η of n−r variables, such that the map

x = Φ(ξ, η) (3)

is a diffeomorfism on Rn, and the dynamics of η ∈ Rn−r, which is referred toas the “internal dynamics” [19], can be expressed in the following form

η = q(ξ, η) (4)

Note that if r = n there are no internal dynamics, and the system is saidto be “fully linearizable” [19].

Let yR be the desired output response, and consider the error dynamics

e(r) = Lrfh(x) − y

(r)R + LgL

r−1f h(x)u (5)

where e = ξ − ξR and ξR = [yR, yR, . . . , y(n−r)R ].

Let us make the following assumption:

Assumption: The internal dynamics

η = q(ξ, η) (6)

is input-to-state stable (ISS) (i.e., for any bounded ξ(t) the internal state η(t)remains bounded irrespectively of the initial conditions.)

85Higher-Order Sliding Modes for the Output-Feedback Control

Let the control task be to solve the finite-time output-feedback trackingcontrol problem.

As usual in the context of the sliding mode approach, a solution to thisproblem is generally characterized by a two-step procedure:

STEP 1. Identify a function s = s(e, e, . . . , e(p)), p ≤ r − 1, called constraint(or sliding) quantity such that any motion of the system on the manifolds = 0 is characterized by the finite-time zeroing of e, e, . . ., e(p).

Possible choices for the sliding quantity s are mentioned in Sect. 5.1 - 5.3.

STEP 2. Find a control u which stabilizes the nonlinear uncertain dynamics(differential inclusion)

s(r−p) = ∂s(r−p−1)

∂e(r−1) [Lrfh(x) − y

(r)R + LgL

r−1f h(x)u] =

= ϕ(ξ, ξR, η) + γ(ξ, η)u(7)

on the basis of suitable known upperbounds to the uncertainties appearingin (7) without requiring knowledge (or estimation) of s, s, . . ., s(r−p−1)

We assume that the uncertainties ϕ and γ globally satisfy the boundednessconditions

|ϕ(ξ, ξR, η)| ≤ F (ξ, ξR)

0 < Γ1 ≤ γ(ξ, η) ≤ Γ2

(8)

where F (·) is a known positive function and Γ1, Γ2 are known constants.

The second step involves the problem of finite-time stabilization of anuncertain differential inclusion of order (r − p) (see Sect. 5, 6).

It must be stressed that while the simple discontinuous control on s = 0is effective if r−p = 1, it is possibly unstable if r−p = 2 and always unstableif r−p > 2 [1]. Therefore, very complex switching logic must be adopted (e.g.the time optimal bang-bang control) considering, furthermore, the difficultiesof predicting the behaviour of nonlinear systems of high order. Actually, step2 represents a formidable research task, especially if r − p > 2.

But what are the pros of incrementing r−p ? In particular, in what senser − p = 2 is better than r − p = 1?

In the following section a preliminary attempt to give an answer to thesequestions is presented through simulations.

3 Motivating examples

The actual research state seems to suggest that the output-feedback controlproblem has been satisfactorily solved for wide classes of systems of practical


relevance. The key tool is to implement an HGO of order p = r−1 and, then,to use it in conjunction with proper control techniques for relative-degree onesystems.

An obvious question which should be addressed is: what is the cost ofhigh-gain observers ?

The parameter affecting the performance of these systems is the observergain, which positively affects two basic requirements: the precision and thepossibility of recovering the region of attraction achieved under state feedback[2]. The general effect of increasing the gain is: the higher the gain, the betterthe overall performance.

While the peaking phenomenon affecting the observer transient can beeasily managed by saturating the system input outside a suitable domain ofinterest, the only cost to pay, as the differentiation order increases, appearsto be the geometric growth of the coefficients of the observer state equa-tion, with the associated stability problems arising from the discrete-timeimplementation.

These facts, seemingly, prevent any attempt to make research relevantto the control of systems with relative degree higher than one; any system,indeed, could be reduced to a relative degree one system thanks to the useof an high gain observer of suitable order.

However, it is well known that a feedback cost can be associated to theeffect of the measurement errors on the plant control input [18]. The resultingphase-distortion and amplification of the system control input strongly affectthe system behaviour.

Anyone can observe that, using an HGO, the frequency response of the p-th order estimated derivative increases with rate 6p dB/octave in a frequencyrange which goes larger and larger as the observer gain increases, with the riskof including the noise spectrum range. Therefore, both the actual gain andthe order of the observer strongly affect the discrepancy between the “ideal”controller and the real one, worsening precision, transient performance andbasin of attraction.

As an example, consider the output-feedback stabilization problem for thefollowing open-loop unstable plant

x1 = x2

x2 = x3

x3 = −2x1 − x2 + x3 + 5 + sin(4t) + u = ϕ(x, t) + u(9)

with output y = x1.Since the input-output relative-degree equals the system order, there are

no internal dynamics. No a-priori information about the drift term ϕ(x, t)is available, besides its upper boundedness by a known function with lineargrowth w.r.t. the state norm

|ϕ(x, t)| ≤ N + M‖x‖1 N = 6 M = 2 (10)


0 5 10 15

0

0.5

1

Time [sec]

y(t)

5 10 150

0.01

0.02

0.03

0.04

0.05

Time [sec]

y(t)

k=50 k=70

k=100

Fig. 1. Noise-free output measurement. The output behaviour

The following relative-degree-one sliding output can be defined

s = y + 4y + 4y (11)

whose coefficients assign a twin pole in −2 to the reduced-order sliding modedynamics.

The first and second output derivatives have been estimated by meansof an HGO with triple pole in −k, k being the observer gain. Then, theconventional relay-type first-order SMC has been implemented.

In Fig. 1 the output responses for k = 50, k = 70 and k = 100 are depicted.It can be seen that the steady-state accuracy goes greater and greater as thegain k increases.

This qualitative behaviour reverses in the presence of noise. Let the mea-sured output be corrupted by additive high frequency noise

N(t) = 0.002sin(200t) (12)

Now, the larger the gain the worse the accuracy (Fig. 2), since the effectof noise amplification predominates on the accuracy improvement due to theincrease of the observer gain.

Remark: Note that low-pass filtering the output before the differentiationattenuates the effect of noise but does not change the qualitative dependenceof the overall accuracy on gain k. Moreover, stability problems may occur inthe closed loop, especially for unstable plants, due to the increase of relativedegree.

On the basis of the above considerations, the order of the observer appearsto be a good candidate to represent the cost associated with the use of highgain observers.


0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

Time [sec]

y(t)

k=50 k=70

k=100

Fig. 2. The effect of high-frequency noise affecting the output

A sensible way to reduce the number of output derivatives required forfeedback (i.e. the observer order) is to use a different type of control algo-rithm, especially devoted to deal with systems with relative degree greaterthan one.

The use of a second-order sliding mode control (2-SMC) algorithm [4,24]is a way for steering to zero the output of a relative-degree-two system bydiscontinously modifying its second derivative with no information demandabout the first derivative. This latter property is crucial, since it means thatthe order of the observer can be reduced by one.

To be more specific, as far as the stabilization problem for system (9) isconcerned, the use of a 2-SMC algorithm allows us to define a relative-degreetwo sliding output of the type

s = y + cy c > 0 (13)

with the apparent benefit that only y needs to be estimated, thus, a first-orderhigh-gain observer can be actually implemented.

Further improvements can be attained if other types of real-time differen-tiation units are considered. Indeed, an important application of 2-SM algo-rithms is the possibility of designing real-time robust differentiators [7,26,28]with a more profitable compromise between accuracy and noise-immunity, ascompared with conventional 1-SM based differentiators [51] and HGOs.

Let us introduce a compact terminology :k-SMC: k-th order sliding-mode controllerHGO-p: HGO for estimating output derivatives up to the p-th order one

In Fig. 3 the results obtained in the three cases:

• 1-SMC combined with HGO-2 (sliding quantity (11)).• 2-SMC combined with HGO-1 (sliding quantity (13)).


0 2 4 6 8 10 12

0

0.5

1

Time [sec]

y(t)

5 6 7 8 9 10 11 12−0.1

0

0.1

0.2

Time [sec]

y(t)

1−SMC / HGO −2

2−SMC / HGO −1

2−SMC / 2−SMC differentiator

Fig. 3. Comparison between 1-SMC and 2-SMC based schemes

• 2-SMC combined with 2-SMC differentiator (sliding quantity (13)).

are depicted

The improvement obtained through the combined use of a 2-SMC and anHGO-1, first, and of a 2-SMC and a 2-SMC differentiator is apparent (Fig.3). A detailed comparative analysis of different solutions and approaches isleft to future research; the aim of the present chapter is mainly pointing outthat there are important unsolved problems in the output feedback controlof nonlinear uncertain systems, and that higher-order sliding modes mayconstitute an effective tool to satisfactorily address such problems.

4 Higher order sliding modes: basic definitions

Let us first remember that according to the definition by Filippov [15] anydiscontinuous differential equation x = v(x), where x ∈ Rn and v(·) is alocally bounded measurable vector function, is replaced by an equivalentdifferential inclusion x ∈ V (x).

In the simplest case, when v(·) is continuous almost everywhere, V (x) isthe convex closure of the set of all possible limits of v(y) as y → x, while yare continuity points of v(·). Any solution of the equation is defined as anabsolutely continuous function x(t) satisfying the differential inclusion almosteverywhere.

For simplicity we restrict ourselves to sliding modes with respect to scalarconstraint functions. General definitions of HOSMs with respect to vector-functions, and of HOSMs on manifolds (see [24,25]) are not considered here.


Let a constraint be given by an equation s(x) = 0, where s : Rn → Ris a sufficiently smooth constraint function. It is supposed that total timederivatives s, s, s, ..., s(k−1) along the system trajectories exist and are single-valued functions of x, which is not trivial for discontinuous dynamic systems.In other words, discontinuity does not appear in the first k−1 total derivativesof the constraint function s.

Then, the k-th order sliding set (k-sliding set) is determined by the equal-ities

s = s = s = ... = s(k−1) = 0 (14)

forming a k-dimensional condition on the state of the dynamic system.

Definition 1. Let the k-sliding set (14) be non-empty and assume thatit is locally an integral set in Filippov’s sense (i.e. it consists of Filippov’strajectories of the discontinuous dynamic system). Then the correspondingmotion satisfying (14) is called k-sliding mode (k-SM) with respect to theconstraint function s.

A sliding mode is called stable if the corresponding integral sliding set isstable. A typical trajectory when approaching a 2-SM is shown in Fig. 4.

0=s

0=s

0== ss

Fig. 4. 2-sliding mode trajectories: the reaching phase

Under the assumption that s, s, s, ..., s(k−1) are differentiable functionsof x, the additional k-sliding regularity condition

rank[∇s,∇s, . . . ,∇s(k−1)

]= k (15)

where ∇ represents the gradient operator, implies that the k-sliding set is adifferentiable manifold and that s, s, s, ..., s(k−1) may be supplemented up tonew local coordinates.


Remarks.

1. It is frequently heard that higher order sliding modes differ in thenumber of successive total derivatives of s which vanish in the sliding modes ≡ 0. Nevertheless, that number cannot be considered as a characteristic ofthe mode, since formally derivatives of any order are nullified. Sliding modesshould be distinguished by system properties outside of them. The most nat-ural characteristic of a sliding mode is the number of successive continuoustotal derivatives of s in a vicinity of the mode. In other words the number kis taken corresponding to the first derivative s(k) which is discontinuous ordoes not exist due to some reason, like trajectory nonuniqueness. k is calledthe sliding order. It characterizes the system motion with respect to the givensliding output s, and may change if another output is considered with thesame sliding motion.

2. The above definitions are easily extended to include non-autonomousdifferential equations (by introduction of the fictitious equation t = 1) andto the case of the closed-loop controlled system x = f(t, x, u), u = U(t, x),with discontinuous U(·) and smooth f(·), s(·).

4.1 Real sliding

Only ideal sliding modes were considered in Def. 1, keeping exactly the k-sliding mode condition (14). However, ideal sliding is achieved by means of acontrol signal commuting at infinite frequency, which cannot be attained inreal plants due to switching imperfections (the simplest switching imperfec-tion is the switching delay caused by discrete measurements). It was proved[24] that the best possible sliding accuracy attainable with discrete switchingin s(k) is

|s|(j) ≈ T k−j j = 0, 1, ...., k (16)

where T > 0 is the minimal switching time interval and s(0) = s.Thus, in order to achieve the k-th order of accuracy in discrete realization,

the sliding order has to be at least k. It is known that standard slidingmodes provide for first-order real sliding only. Real sliding of higher ordersare achieved by discrete switching modifications of higher order sliding modeswith finite-time convergence [31]. Real sliding of second and third orders areobtained by special discrete switching algorithms (see Sect. 6.2) [10,9,24,42].

5 Higher Order Sliding modes with finite-timeconvergence

Actually, HOSMs definitions in the previous Section leave out the problem ofhow such kinds of motion are established, simply defining them as a specialtype of sliding mode that satisfies the set of constraints (14).


Note that asymptotically stable, or unstable, HOSMs may appear in vari-able structure systems with fast actuators, as it was pointed out in [25,22]. Inthat case, stable HOSMs reveal themselves by spontaneous disappearance ofthe chattering effect. Dynamical sliding mode control [41,40] leads to asymp-totically stable higher-order sliding modes, and has to be specially mentionedhere.

5.1 Terminal sliding modes

An interesting family of sliding surfaces featuring finite-time convergence isused to achieve the so-called “terminal sliding modes” [50].

Terminal sliding surface corresponds to the the vanishing of a slidingvariable str

which is defined by means of an iterative procedure:

st0 = e (17)sti

= sti−1 + ci sti−1pi/qi i = 1, 2, . . . , r − 1 (18)

str= str−1 + cr−1 str−1

pr/qr (19)

Here pi, qi (pi < qi) are suitable integer odd coefficients [50]. Sufficientconditions for the stability of a discontinous feedback

u = −Usign(str) (20)

are derived in [50] for a given class of plants, assuming full-state availabilityfor the real-time feedback calculation.

Unfortunately, the resulting closed-loop system may have an unboundedright-hand side (for some initial conditions) which prevents the very imple-mentation of the Filippov theory for this situation. The corresponding controlis formally bounded along each switching manifold, but may take infinite val-ues in some vicinity of the sliding manifold [52].

5.2 Arbitrary-Order Sliding Controllers

Another class of sliding controllers is based on arbitrary-order sliding modeswith finite time convergence [31].

Let us recall that r is the relative degree of system (1), and let m beany positive number, the common choice being the least common multiple of1, 2, . . . , r.

Define the following quantities

N1,r = |e|(r−1)/r (21)

Ni,r =(|e|m/r + |e|m/(r−1) + . . . + |e(i−1)|m/(r−i+1)

)(r−i)/m

(22)

i = 1, . . . , p− 2

Nr−1,r =(|e|m/r + |e|m/(r−1) + . . . + |e(r−2)|m/2

)1/m

(23)


and consider also

φ0,r = e (24)φ1,r = e + β1N1,r sign(φ0,r) (25)

φi,r = e(i) + βiNi,r sign(φi−1,r) i = 2, . . . , r − 1 (26)

The following Theorem was proved in [31]:

Theorem 1. Consider system (1), (8) with F (·) ≡ F = const. Assume thattrajectories of system (1) are infinitely extendible in time for any Lebesgue-measurable bounded feedback control. Then, with properly chosen positive pa-rameters β1, β1, ..., βr−1, α, the controller

u = −α sign(φr−1,r(e, e, ..., e(r−1))) (27)

where Φr−1,r(·) is defined in (21)–(26) leads to the establishment of a finite-time converging r-sliding mode on the manifold e = 0, and the resultingclosed-loop system is finite-time output-stable.

The above Theorem determines a controller family applicable to all sys-tems of the type (1) with relative degree r, satisfying (8), with F (·) ≡ F ,for some constants F, Γ1, Γ2. Parameters β1, ..., βr−1 affect the reaching timeand are to be chosen sufficiently large in index order. While the number ofchoices of βi is certainly infinite, it is possible to take some predefined valueschosen for each r in advance. Parameter α > 0 must be chosen specifically forany fixed F, Γ1, Γ2. As the value of α is determined by the upper bounds ofthe input-output dynamics, the controller performance is insensitive to anysystem perturbation preserving these bounds.

Following are a few examples of sliding controllers with βi tested for r ≤3,m being the least common multiple of 1, 2, ..., r.

1. u = −α sign e2. u = −α sign(e + |e|1/2sign e)3. u = −α sign(e + 2(|e|3 + |e|2)1/6sign(e + |e|2/3sign e)4. u = −α signe(3) + 3(s6 + e4 + |e|3)1/12sign[s+

(e4 + |e|3)1/6sign(e + 0.5|e|3/4sign e)]

(28)

The rationale of the controller is that a 1-sliding mode is established inthe continuity points of the discontinuity set

Γ : φr−1,r = 0 (29)

Such a sliding mode is described by the differential equation φr−1,r = 0 pro-viding in its turn for the existence of a 1-sliding mode φr−2,r = 0. Note thatthe primary sliding mode disappears when the secondary one is to appear.


The resulting movement takes place in some vicinity of the subset of Γ sat-isfying ψr−2,r = 0, then it transfers in finite time into some vicinity of thesubset satisfying ψr−3,r = 0 and so on. While the trajectory approaches ther-sliding set, set Γ contracts to the origin in the coordinates e, e, ..., e(r−1).

Controller (27) requires the availability of e, e, ..., e(r−1). Note that e(r) isbounded, which provides for the possibility of using an (r−1)-th order exactdifferentiator in the feedback (see Sect. 7). The information demand may alsobe lowered by use of a suitable discretized controller [31]:

Theorem 2. [31] Consider system (1) satisfying the same assumptions anddefinitions as in Theorem 1, and let the measurements be carried out at thesampling instants ti, with constant step T > 0. Then, with properly chosenpositive parameters β0, β1, ..., βr−1, α the controller

u(t) = ui = −αsign(∆ie

(r−2) + βiTNr−1,rsign(φr−2,r

(ei, ei, ..., e

(r−2)i

)))

ti ≤ t < ti+1

(30)

where ei = e(ti), ei = e(ti), . . . , e(r−2)i = e(r−2)(ti), and ∆ie

(r−2) =e(r−2)i − e

(r−2)i−1 , provides for the finite-time fulfillment of inequalities

|e| ≤ a0Tr , |e| ≤ a1T

r−1 , . . . , |e(r−1)| ≤ ar−1T (31)

where aj, j = 0, 2, . . . , r− 1 are some positive constants and the convergencetime is a locally-bounded function of initial conditions.

That is the best possible accuracy attainable with discontinuous e(r) sep-arated from zero [24]. The discontinuous manifold φr−1,r may be replacedby its smooth approximation by means of some regularization procedure[27]. A model example, and relevant computer simulations, can be foundin [31,27,28].

Some additional remarks on the use of the proposed controllers follow.

Remarks

1. Implementation of r-sliding controller when the relative degree is less thanr (chattering avoidance).

Introducing successive time derivatives u, u, ..., u(r−k−1) as new auxiliaryvariables, and u(r−k) as a new control (dynamical extension), one achievesdifferent modifications of each r-sliding controller devoted to control systemswith relative degrees k = 1, 2, ..., r. The resulting control input, obtained atthe output of a chain of (r − k) integrators, turns out to be an (r − k − 1)-smooth function of time with k < r, a Lipschitz function with k = r − 1,and a bounded “infinite-frequency switching” function with k = r. Using thesame trick the chattering effect can be removed.

2. Controlling systems not affine on control.


If the system is nonlinear on control, one can differentiate the input-output dynamics until a system affine in the higher-order control derivativeis obtained. The problem may be solved, having introduced as before newauxiliary state variables u, u, ...,, using the higher-order control time deriva-tive as the new control variable.

5.3 Time-Optimal Sliding Surfaces

The time optimal bang-bang control law represents, historically, the first ex-ample of higher order sliding mode with finite time convergence. The strategydeveloped for double and triple integrator can be expressed in the form of adiscontinuous control across a switching surface [34,12].

Any uncertain system of the same order (2 or 3) forced to evolve on theassociated time-optimal surface, e.g. by means of first-order sliding modecontrol, results in being characterized by the same finite-time convergenceproperty [34,12].

Sliding outputs defining surfaces of order 2 and 3 are

sTO2 = e + ke|e| k > 0 (32)

sTO3 = e+13e−

[ee +

(12e2 + e sign

(e +

12e|e|

))]sign

(e +

12e|e|

)(33)

Note that with a first order sliding mode controller the system can beforced to perform a higher order sliding motion.

6 Recent results on 2-SMC design

Previous considerations regarding the positive implications of taking r − p(the relative degree between the constraint variable and the discontinuouscontrol) as high as possible (see Section 2) can justify the research activ-ity on higher-order sliding controllers. While the previous Section 5 was de-voted to illustrate some possible choices for sliding manifolds with finite-timeconvergence, the aim of the present Section is to give an overview aboutrecently-developed methodologies to enforce a second-order sliding mode onthe chosen sliding manifold.

Up to now only few 2-sliding controllers have been proposed [4]. The so-called “super-twisting” algorithm is conceptually different from the others, fortwo reasons: first, it depends only on the actual value of the sliding constraint,while the others have more information demands. Second, it is effective onlyfor anti-chattering purposes as far as relative-degree one constraint variablesare dealt with.


On the contrary, both the “twisting” and the “sub-optimal” algorithm [4]can deal with relative-degree two constraint variables. They are, both, specialcases of the general algorithm

u(t) = −α(s, sM )UM sign(s− βsM )

α(s, sM ) =

1 if(s− βsM )sM ≤ 0α∗ otherwise

(34)

where sM is the last (meaning the most recent) point with a zero derivative(“singular point”) of s, UM > 0, and α, β ∈ (0, 1], are suitable coefficientsto be set, according to the uncertainty bounds, in order to guarantee theexistence and stability of the 2-SM. Note that β = 0 and β = 1/2 for the“twisting” and the “sub-optimal” algorithms, respectively.

The recently proposed 2-SMC with global convergence features [11], forwhich α = 1 and β is adaptively adjusted on-line, belongs to the above classof sliding controllers.

The distinguishing feature of such class of controllers is that the controllaw depends on the current value of the sliding constraint and on its pasthistory, represented by the last occurred singular point sM .

While these algorithms are actually implemented looking at the first dif-ference of s (whose sign approximates the sign of s), a different controllerstructure, which stores and processes the past time history of s with no at-tempt to estimate the sign of s, is also possible.

The accuracy of these algorithms, both in their continuous and discrete-time implementation, can be improved by means of proper learning and adap-tation procedures, which counteract the chattering effect as well [6,3,10,9].

6.1 Second-Order Sliding Mode Controllers with GlobalConvergence

As far as uncertain systems confined in a known bounded domain are con-cerned, it is possible to relate a-priori the control amplitude and the switchinglogic to the magnitude of the uncertainties, such that the transient is stableand finite-time converging.

In a recent work this assumption has been dispensed with, and a globally-converging algorithm has been presented [11]. To this end, a crucial role hasbeen played by one of the parameters affecting the transient of second orderdiscontinuous differential equations, i.e. the so-called “anticipating factor” βin (34) [4].

The following Theorem 3 illustrates the proposed controller; then a simplesimulation example is provided in order to highlight its good features.


Theorem 3. Consider system (1) with r = n and satisfying (8) withF (ξ, ξR) ≤ F (s, s), F (s, s) being a known function. Let s(t) = s(e, e, . . . ,e(r−2)) be the scalar constraint quantity. Then, the control law

u(t) =− 1

Γ1[F [ξ(0)] + χ] sign (s(0)) t = 0

− 1Γ1

[F [ξ(t)] + χ] sign (s(t) − s(0)) 0 < t ≤ tM1

χ > 0 (35)

ensures the finite-time reaching of a first singular point sM1 . From this pointon, the control law

u(t) = −UMksign [s(t) − βksMk

] tMk< t ≤ tMk+1 k = 1, 2, . . . (36)

guarantees the global finite-time vanishing of the constraint variable s andof its unmeasurable derivative s, provided that the controller parameters arechosen according to

UMk=

α

Γ1

[F

(sMk

, η√

|sMk|)

+13η2

]α > 1 (37)

βk = max

12, 1 − η2

2[F(sMk

, η√|sMk

|)

+ Γ2UMk]

(38)

where η is a positive constant, tMkare the time instants at which s is zero,

and sMk= s(tMk

).

An interesting feature of the above algorithm is that, due to the adaptiveswitching rule, it allows for counteracting the transient peaking and it iseffective also when the controlled system may exhibit finite escape time.

As an example, consider the output stabilization problem for the system

y = ay2 + u |a| ≤ 4 (39)

with unmodelled actuator

τ u = −u + v τ > 0 (40)

The input-output relative degree is two, and y is not measurable. TheHGO-based 1-SMC scheme, with s = ˆy + y, and the above presented global2-SMC scheme, with s = y, are compared. Note that using 2-SMC no out-put differentiation is required. The significant counteraction of the peakingphenomenon is apparent from Fig. 5.

6.2 Second-Order Sliding Mode Control for sampled-datasystems

Consider system (1) under the same assumptions as in Theorem 1, and letsi be the sequence of sampled values of the sliding quantity, si = s(iT )


0 1 2 3

-1

0

1

2

Time [sec] Time [sec]

HGO-2 / 1-SMC Global 2-SMC

Fig. 5. Combined HGO-1/1-SMC scheme and global 2-SMC scheme. A comparisonof the output response.

(i = 0, 1, . . .), T being the sampling period. The plant input is piecewise-constant (ZOH device), i.e., u(t) = ui, t ∈ [i T, (i + 1) T ).

The discrete-time version of the sub-optimal 2-SMC algorithm can bederived by direct discretization of the continuous-time algorithm, providedthat the control amplitude is properly set [4], that is

ui = −αiUM sign[si − 1

2sMj

](41)

UM ∈(

F

α∗Γ1,∞

)∩

(4F

3Γ1 − α∗Γ2+ θ1T, θ2T

−2

)

(42)

where θ1, θ2 are proper constants, sMjis an estimate of the last singular value

of s (see (34)), obtained by the following approximate digital peak-detector:

set s−1 = s(0) ; s−2 = 0 j = −1set Λi = (si − si−1)(si−1 − si−2)

If (Λi ≤ 0) then

j = j + 1sMj

= si−1

(43)

and αi is adjusted according to

αi =

1 ifsi − 1

2 sMj

sMj

< 0α∗ otherwise

(44)

with the constant α∗ ∈ (0, 1) ∩(

0, 3Γ1Γ2

).


It has been proven [5] that the above digital control strategy guaranteesthe reaching of the boundary layer (16) of the 2-sliding set s = s = 0 after afinite number of sampling instants. However, controller (41)–(44) reveals analmost ringing behaviour within the boundary layer.

Nevertheless, assuming that the uncertainties affecting the sliding variabledynamics (7) are locally Lipschitz, it was shown in [5] that the sliding variabledynamics can be represented, with an O(T 3) approximation, by the followingdiscrete model:

si+1 = 2si − si−1 + ϕi−1T2 + 1

2 (γiui + γi−1ui−1)T 2 (45)

By resorting to the extension of the equivalent control concept to thediscrete-time setting [47,48], the corresponding discrete–time equivalent con-trol (DTEC) can be defined as

udeqi

= − 1γi

(γi−1ui−1 + di + 22si−si−1

T 2

)(46)

where di is unknown due to system uncertainties. Using a one step delayestimate of di, computed by means of discrete model (45), the DTEC can beapproximated as follows

udeqi

= 1γn

(γnui−2 − 2 3si−3si−1+si−2

T 2

)(47)

where γn =√Γ1Γ2 is a reasonable estimate of the uncertain control gain.

The main problem in using the DTEC method is that the amplitude ofthe DTEC is proportional to s[k]/T as far as systems with relative degreeone are dealt with [42], and to s[k]/T 2 in the considered case [5]. Therefore,in the case under investigation, a O(T 2)-vicinity of the sliding manifold mustbe achieved first. Next application of control law (47) provides the finite-timeattainment of a O(T 3) sliding accuracy, which is the highest achievable onewhen a piecewise-constant control signal forces to zero a relative-degree-twooutput variable. That is the same accuracy of real 3-SM.

The presence of uncertainties in control gain makes stability analysisrather involved [42]. Suitable assumptions regarding the uncertainties areneeded to ensure that the system trajectory reaches, and does not leave,the O(T 3) boundary layer of s = 0. A qualitative requirement is that theuncertainty in the control gain is “sufficiently small”.

The following Theorem was proved in [9]:

Theorem 4. Consider system (1) with the same assumptions as in Theorem1. Assume the directly discretized control (41)–(44) has driven the system inan O(T 2) vicinity of s = 0. The successive application of the control law

ui = udeqi

(48)


where udeqi

is defined in (47), guarantees that the accuracy is improved up to

|s(t)| ≤ O(T 3)t ≥ T ∗

|s(t)| ≤ O(T 2)(49)

T ∗ being a finite transient.

7 Real-time output differentiation via sliding modes

In this section a sliding-mode solution to the multiple differentiation problemis reported, and a particular emphasis is devoted to the recently presentedarbitrary-order finite-time converging robust differentiator ([28]).

Let a Lebesgue-measurable locally bounded input signal h(t) be definedon [0,∞) and let it consist of an unknown smooth base signal h0(t) and anunknown-but-bounded Lebesgue-measurable noise N(t) with sup |N(t)| = ε,ε being unknown as well. Let the p-th derivative of h0(t) have a knownLipschitz constant Cp.

7.1 First-order SM differentiators

The first problem is to find an estimation of h0(t), robust in presence of N(t)and exact when N(t) = 0.

The conventional differentiator based on first-order sliding modes takesthe form:

ζ = vv = −µ sign(ζ − h(t))τ vav + vav = v

(50)

where µ > C1, τ > 0, C1 = max |h0|. According to [47,51], vav is a O(τ)-estimate of the derivative of h. Cascade implementation is feasible, but eachstage increases noise sensitivity.

Under the above smoothness assumptions, it has been shown in [24] thatno differentiator can provide for differentiation accuracy of the j-th derivative(j = 0, 1, ..., p) better than C

j/pp ε(p−j)/p. It has been also proved in [24] that

there is a differentiator providing for the accuracy proportional to the just re-ported one, but, unfortunately, that was only a pure existence theorem. Notethat if additional restrictions are imposed on h0 the adduced performancemay be, in principle, improved.

It should be stressed that, due to the averaging, differentiator (50) has anintrinsic error also in the absence of noise, thus it is robust but not exact.

To overcame the need to known a Lipschitz constant of the signal to dif-ferentiate, a combined adaptive-variable structure scheme has been proposed


in [49]. A dead-zone based adaptation mechanism give the proposed schemerobustness against the measurement noise. Similarly to the conventional 1-SMC differentiator (50), the averaging of the output makes the scheme in[49] robust but not exact.

On the contrary, an exact and robust 2-SM first-order differentiator canbe based on the “super-twisting” algorithm [26]:

ζ(t) = v(t)v(t) = θ(t) − λ|ζ(t) − h(t)|1/2 sign(ζ(t) − h(t))θ(t) = −µ sign(ζ − h(t))

(51)

Here λ, µ > 0, and both v(t) and θ(t) may be considered as the outputof the differentiator. Solutions of the system are understood in the Filippovsense.

Parameters may be chosen, for example, in the form µ = 1.1C2, λ =1.5C1/2

2 , where C2 = max |h0|. Other possible criteria are given in [26]. Dif-ferentiator (51) provides for finite-time convergence to the exact derivativeof h0(t) if N(t) = 0. Otherwise, it provides for accuracy proportional toC

1/22 ε1/2, which is the best possible precision in the considered case [26].

In case of a p-stage cascade implementation, differentiator (51) will pro-vide for p-th order differentiation accuracy of the order of ε2

−p

. All necessaryconditions for successive p-th order differentiation, and implementation ofhigher-order sliding controller (Sect. 5.2), are fulfilled, at least locally, forsmooth dynamic systems with relative degree p. Thus, full local real-time ro-bust control of output variables is possible, using only output variable mea-surements and knowledge of the relative degree. Nevertheless, the cascade ofdifferentiation stages is not satisfactory for multiple differentiation purposes,due to excessive noise propagation. It is more effective to implement higher-order differentiators, specially designed for the multiple differentiation task,which are described in the following subsection.

7.2 Arbitrary-order, exact, finite-time converging, robustdifferentiation

The aim is now to find real-time robust estimates of h0(t), h0(t), ..., h(p)0 (t),

exact in the absence of measurement noise and continuously depending onits magnitude (e.g. robust). A recursive design scheme is proposed [32].

Let a (p − 1)-th-order differentiator Dp−1(h(t), Cp−1) produce outputsDi

p−1 (i = 0, 1, ..., p − 1) which are estimates of h0, h0, , ..., h(p−1)0 for any

input signal h with h(p−1)0 having Lipschitz constant Cp > 0.


Then, the p-th order differentiator has the outputs zi = Dip , i = 0, 1, ..., p,

defined as follows:

z0 = ν, ν = −λ|z0 − h(t)| pp+1 sign(z0 − h(t)) + z1,

z1 = D0p−1(ν, Cp), . . . , zp = Dp−1

p−1(ν, Cp)(52)

Here D0(h(t), Cp) is a simple nonlinear filter

D0 : z = −λ sign(z − h(t)) , λ > Cp. (53)

In other words it has the form

z0 = ν0, ν0 = −λ0|z0 − h(t)| pp+1 sign(z0 − h(t)) + z1,

. . .

zi = νi, νi = −λi|zi − νi−1|p−i

p−i+1 sign(zi − νi−1) + zi+1,. . .zp = −λp sign(zp − νp−1)

(54)

It is easy to check that the above-presented p-th order differentiator canbe expressed in the following non-recursive form:

z0 = z1 − κ0|z0 − h(t)| pp+1 sign(z0 − h(t))

z1 = z2 − κ1|z0 − h(t)| p−1p+1 sign(z0 − h(t))

. . .

zi = zi − κi|z0 − h(t)| p−ip+1 sign(z0 − h(t))

. . .zp = −κp sign(z0 − h(t))

(55)

for suitable positive constant coefficients κi.Admissible values for the coefficients in (54) are easier to find than those

in (55). In fact, in the first case, the values λ0, . . . , λp−1 defining the (p−1)-thorder differentiator still apply for the p-th order one, and, therefore, one moreparameter needs to be evaluated. See [32] for more details on the parametersdesign procedure.

Finite-time convergence and Lyapunov stability as well are proved in [32]on the basis of the concept of homogeneous vector fields [36]. In [32] a dif-ferent procedure for building another class of differentiators enjoying similarconvergence properties has been also proposed.

In the following Theorems 5 and 6 the performance of the proposed dif-ferentiator class in presence of bounded measurement noise are investigated,together with discrete-time implementation, respectively.

Theorem 5. Let the input noise satisfy the inequality |h(t) − h0(t)| ≤ ε.Then the following inequalities are fulfilled in finite time for some positiveconstants µi, νi depending only on the parameters of differentiator (55)

|zi − h(i)0 (t)| ≤ µi ε

(p−i+1)(p+1) i = 0, ..., p

|vi − h(i+1)0 (t)| ≤ νi ε

(p−i)(p+1) i = 0, ..., p− 1

(56)


Using recursive high-order differentiators, noise propagation is better coun-teracted than with the cascade implementation of first-order differentiators(Sect. 7.1).

Consider the discrete-sampling case, where z0−h(t) is replaced by z0(tj)−h(tj), with tj ≤ t < tj+1, tj+1 − tj = T > 0.

Theorem 6. Let differentiator (55) be implemented via Eulero discretizationwith the sampling period T > 0, and let the measurements of h be free of noise.Then the following inequalities are fulfilled in finite time for some positiveconstants µi, νi depending exclusively on the parameters of the differentiator

|zi − h(i)0 (t)| ≤ µi T

p−i+1 , i = 0, ..., p

|vi − h(i+1)0 (t)| ≤ νi T

p−i , i = 0, ..., p− 1(57)

Combining controller (27), with p = r− 1, and the (r− 1)-th order exactdifferentiator (54), the tracking problem is ideally solved in finite time underthe conditions in Theorem 1. A model example, and simulation results, arereported in [27,28].

7.2.1 Simulation examples of arbitrary-order differentiation

Differentiator (54) of order 5 with C5 = 1 and coefficients λ0 = 50, λ1 = 30,λ2 = 16, λ3 = 8, λ4 = 4, λ5 = 2.2 has been tested. These parameterscan be easily changed, since the differentiator is not very sensitive to thesevalues. The tradeoffs are as follows: the larger the parameters, the faster theconvergence and the higher the sensitivity to input noise and discretization.The estimation of the i-th derivative achieved by means of the k-th orderdifferentiator is denoted as Di

k(t).Third-order differentiation of the noise-free input signal

f(t) = f0(t) = sin0.5t + cost. (58)

was first considered. Mutual graphics of f(t) and D13, f(t) and D2

3, and,finally, f (3)(t) and D3

3 with the measurement step τ = 5 · 10−5 are shownin Fig. 6.a. As for the fifth-order differentiator, pictures of h(j)(t) and Dj

3,j = 1, 2, ..., 5 are depicted in Fig. 6.b.

The attained accuracies are 1.46·10−13, 7.16·10−10, 9.86·10−7, 3.76·10−4,0.0306 and 0.449 for tracking the signal and the first, second, third, fourthand fifth derivatives, respectively. Reducing τ accuracy is improved accordingto Theorem 6.

Third-order differentiation results are considered in Fig. 6c in the presenceof a periodic non-differentiable high-frequency noise signal with maximummagnitude 10−4.

Estimates of the first and second derivatives produced by the 5th orderdifferentiator, with noise magnitude 0.01 and frequency about 10000 Hz, are


Fig. 6. High-order differentiation

shown in Fig. 6.d. The highest derivative has large phase-deviation, but donot chatter much. The accuracies of estimations D1

5, D25, D3

5, D45 and D5

5 are0.0002, 0.0136, 0.184, 0.649, and 0.740, respectively. Changing noise magni-tude is consistent with the statement of Theorem 6.

8 Conclusions

In this chapter, following the sliding-mode approach, we have addressed theoutput-feedback control problem of nonlinear uncertain plants, which is one ofthe main topics of modern nonlinear control theory and has a strong impact inmany control applications. In this context, recent developments of the theoryof higher order sliding modes appear to furnish promising alternatives, andpossible complement, to other existing approaches.

While presenting the most recent results of the theory, the authors aimat identifying the role of this particular constrained motion in the frameworkof an output feedback design process. In particular, the three elements whichcharacterize the sliding mode approach (the sliding constraint design, thesynthesis of the discontinuous control law, and the estimation of the required


number of output derivatives) can be endowed with finite-time transient andhigher accuracy when higher-order sliding modes are implemented.

Just to give a perspective to the work done in this chapter, the crucialproblem of the counteraction of the effect of measurement errors has been in-troduced, which could be a reasonable paradigm for the comparison betweenthe various existing solutions. A systematic analysis of the relevant propertiesof control schemes with higher order sliding modes represents a challengingtopic for future investigations.

References

1. Anosov, D.V., (1959) On stability of equilibrium points of relay systems, Au-tomatica i telemechanica (Automation and Remote Control), 2, pp. 135–149

2. Atassi, N.A., and Khalil, H.K., (1999) A Separation Principle for the Stabiliza-tion of a Class of Nonlinear Systems, IEEE Trans. on Aut. Control, 44, pp.1672-1687

3. Bartolini, G., Ferrara, A., Pisano, A., Usai, E., (1998) Adaptive reduction ofthe control effort in chattering free sliding mode control of uncertain nonlinearsystems”, Journal of Applied Mathematics and Computer Science, 8, 1, SpecialIssue “Adaptive Learning and Control using Sliding Modes”, X. Yu ed., pp. 51-71

4. Bartolini, G., Ferrara, A., Levant, A., and Usai, E., (1999) On Second Or-der Sliding Mode Controllers, in “Variable Structure Systems, Sliding Modeand Nonlinear Control”, K.D. Young and U. Ozguner (Eds.), Lecture Notes inControl and Information Sciences, Springer-Verlag, 247, pp. 329-350

5. Bartolini, G., Pisano, A., Usai, E., (1999) Variable Structure Control of Non-linear Sampled Data Systems by Second Order Sliding Modes”, in “VariableStructure Systems, Sliding Mode and Nonlinear Control”, K.D. Young and U.Ozguner (Eds.), Lecture Notes in Control and Information Sciences, Springer-Verlag, 247, pp. 43-68

6. Bartolini, G., Levant, A., Pisano, A., Usai, E. (1999) 2-Sliding Mode withAdaptation, Proceedings of the 7th Mediterranean Conference on Control andAutomation (MED’99), pp. 2421-2429, Haifa, Israel.

7. Bartolini, G., Pisano, A., Usai, E., (2000) First and Second Derivative Estima-tion by Sliding Mode Technique, J. Signal Processing, 4, 2, pp. 167-176

8. Bartolini, G., Levant, A., Pisano, A., Usai, E., (2000) On the robust stabiliza-tion of nonlinear uncertain systems with incomplete state availability, ASMEJ. of Dyn. Syst. Meas. and Contr., 122, pp. 738-745

9. Bartolini, G., Pisano, A., Usai, E., (2000) Digital Sliding Mode Control withO(T 3) accuracy, in “Advances in Variable Structure Systems - Analysis, In-tegration and Application”, proceedings of the 6th IEEE Int. Workshop onVariable Structure Systems, Gold Coast, Australia, December 2000, X.Yu andJ.-X.Xu (Eds.), pp. 103-112, World Scientific Publishing, Singapore

10. Bartolini, G., Pisano, A., Usai, E., (2001) Second-order sliding mode control ofsampled-data systems, Automatica, 37, 9 in press

11. Bartolini, G., Pisano, A., Usai, E., (2001) Global stabilization of nonlinearuncertan systems with unmodelled actuator dynamics, IEEE Trans. on Aut.Control, to appear


12. Bartolini, G., Pillosu, S., Pisano, A., Usai, E., (2001) Time-Optimal Stabiliza-tion for a Third-Order Integrator: a Robust State-Feedback Implementation, inF. Colonius and L. Gruene (Eds.), “Dynamics, Bifurcations and Control”, Lec-ture Notes in Control and Information Sciences, Springer Verlag, Heidelberg,in press

13. Battilotti, S., (2001) A Unifying Framework for the Semi-Global Stabiizationof Nonlinear Uncertain Systems via Measurement Feedback, IEEE Trans. Aut.Contr, 46, 1, pp. 3-16

14. Byrnes, C., and Isidori, A., (2000) Output Regulation for Nonlinear Systems:An overview, Int. J. Rob. Nonl. Contr., 10, 5, pp. 323-337

15. Filippov, A., (1988) Differential Equations with discontinuous right-hand side,Kluwer, Dordrecth, The Netherlands

16. Golembo, B.Z., Emelyanov, S.V., Utkin, V.I., and Shubladze, A.M., (1976) Ap-plication of piecewise-continuous dynamic systems to filtering problems, DANSSSR, 226, 5, 1034-1047 (in Russian)

17. Gopalswamy, S., and Hedrick, K., (1993) Tracking nonlinear nonminimumphase systems using sliding control, Int. J. of Control, 68, pp.1141-1158

18. Horowitz, I., and Sidi, M., (1972) Synthesis of feedback systems with largeplant ignorance for prescribed time-domain tolerances, Int. J. Contr., 16, pp.287-309

19. Isidori, A., (1995) Nonlinear Control Systems. Third edition, Springer Verlag,Berlin

20. Isidori, A., (2000) A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback, IEEE Trans. Aut.Contr., 45, 10, pp. 1817-1827

21. Khalil, H.K., (1996) Nonlinear Systems, Prentice Hall, N.J., Second Edition22. Fridman, L., (1993) Singular extension of the definition of discontinuous sys-

tems and stability, Differential equations, 10, pp. 1307-131223. Krstic, M., Kanellakopoulos, I., and Kokotovic P.V., (1995) Nonlinear and

Adaptive Control Design, New York, Wiley24. Levant, A., (1993) Sliding order and sliding accuracy in sliding mode control,

Int. J. of Control, 58, pp. 1247–126325. Fridman, L., and Levant, A. (1996) Sliding Modes of Higher Order as a Natural

Phenomenon in Control Theory, in Robust Control via Variable Structure &Lyapunov Techniques, Lecture Notes in Control and Information Science, F.Garofalo, L. Glielmo (Eds.), Springer-Verlag, London, 217, pp. 107-133

26. Levant, A., (1998) Robust Exact Differentiation via Sliding Mode TechniqueAutomatica, 34, pp.379–384

27. Levant, A., (1999) Controlling output variables via higher order sliding modes,Proc. of ECC’99, Karlsruhe, Germany.

28. Levant, A., (2000) Higher order sliding: differentiation and black-box control,Proc. of CDC’2000, Sidney, Australia

29. Levant, A., Pridor, A., Gitizadeh, R., Yaesh, I., Ben-Asher, J.Z. (2000) Aircraftpitch control via second-order sliding technique, AIAA Journal of Guidance,Control and Dynamics, 23, 4, pp. 586-594

30. Levant, A., (2000), Variable Measurement step in 2-sliding control, Kybernet-ica, 36, pp. 77-93

31. Levant, A., (2001) Universal SISO sliding-mode controllers with finite-timeconvergence, IEEE Trans. Aut. Contr., to appear


32. Levant, A., (2001) Higher order sliding modes and arbitrary-order exact robustdifferentiation, Proc. of ECC’2001, Porto, Portugal

33. Maggiore, M., (2000) Output Feedback Control: A State-Variable Approach,PhD Dissertation. Ohio State University, Ohio, USA

34. Newman, W.S., (1990) Robust Near Time-Optimal Control, IEEE Trans. Aut.Contr., 35, 7, pp. 841-844

35. Oh, S., and Khalil, H.K., (1997) Nonlinear output feedback tracking usinghigh-gain observers and variable structure control, Automatica, 33, 10, pp.1845-1856

36. Rosier, L., (1992) Homogeneous Lyapunov function for homogeneous continu-ous vector field, Syst. Contr. Lett., 19, 4, pp. 467-473

37. Sepulchre, R., Jankovic M., and Kokotovic, P.V., (1996) Constructive NonlinearControl. New York, Wiley

38. Serrani, A., and Isidori, A., (2000) Global robust output regulation for a classof nonlinear systems. Int. J. Rob. Nonlin. Contr., 10, pp. 133-139

39. Serrani, A., Isidori, A., and Marconi, L., (2000) Semiglobal robust regulation ofminimum phase nonlinear systems. Int. J. Rob. Nonlin. Contr., 10, pp. 379-396

40. Sira-Ramirez, H., (1993) On the dynamical sliding mode control of nonlinearsystems, Int. J. of Control, 57, 5, pp. 1039-1061

41. Spurgeon, S.K., and Lu, X.Y., (1997) Output tracking using dynamic slidingmode techniques Int. J. Robust and Nonlinear Control, Special Issue on NewTrends in Sliding Mode Control, 7, 4, pp. 407-427

42. Su, W.C., Drakunov, S.V., and Ozguner, U., (2000) An O(T 2) Boundary Layerin Sliding Mode for Sampled-Data Systems, IEEE Trans. Aut. Contr., 45, 3,pp. 482-485

43. Teel, A., and Praly, L., (1994) Global stabilizability and observability implysemi-global stabilizability by output feedback, Syst. Contr. Lett., 22, pp. 313-325

44. ———- (1995) Tools for semi-global stabilization via partial state and outputfeedback, SIAM J. Contr. Opt., 33, pp. 1443-1488

45. Tornambe’, A., (1992) High-Gain Observers for non-linear systems, Int. J. Syst.Sci., 23, 9, pp. 1475-1489

46. Tornambe’, A., (1992) Output feedback stabilization of a class of nonminimumphase nonlinear systems, Syst. Contr. Lett., 19, pp. 193-204

47. Utkin, V.I., Sliding Modes in Control and Optimization, Springer Verlag,Berlin, 1992

48. Utkin, V.I., and Drakunov, S.V., (1989) On Discrete–Time Sliding Mode Con-trol”, Proc. of the IFAC Symposium on Nonlinear Control Systems - NOLCOS’89, pp. 484–489, Capri, Italy

49. Xu, J.-X., Jia, Q.-W., and Lee, T.H., (2000) On the Design of a NonlinearAdaptive Variable Structure Derivative Estimator, IEEE Trans. Aut. Contr,45, 5, pp. 1028-1033

50. Yu, X. and Zhihong M. (1996) Model Reference Adaptive Control Systems withTerminal Sliding Modes, Int. J. Contr., 64, pp. 1165-1176

51. Yu, X., Xu, J.X. (1992) Nonlinear derivative estimator, Electronic Letters, 32,pp.16

52. Wu, Y., Yu, X., and Man, Z., (1998) Terminal sliding mode control design foruncertain dynamic systems” Systems and Control Letters, 34, 5, pp. 281-288


Variable Structure Systems with TerminalSliding Modes

Xinghuo Yu1 and Zhihong Man2

1 Faculty of Informatics and Communication, Central Queensland University,Rockhampton QLD 4702, Australia.

2 School of Computer Engineering, Nanyang Technological University, Singapore639798.

Abstract. In this chapter, we discuss recent developments in a special topic ofVariable Structure Systems, namely, the terminal sliding mode control. Dynamicproperties of the terminal sliding mode control systems are explored and theirapplications in single input single output (SISO) systems and multi input multioutput (MIMO) systems are presented. Further improvements of the particularsliding mode control strategy are suggested.

1 Introduction

This chapter discusses recent developments in a special topic of VariableStructure Systems (VSS), that is, the terminal sliding mode (TSM) control.The effectivenss of the VSS approaches has been well documented in theabundant VSS literature as well as the chapters in this book. The robustnessand simplicity for implementation are the trademarks of VSS. The key con-cept in the VSS is the so called sliding mode behavior, which is attained bydesigning the control laws which drive the system state to reach and remainon the intersection of a set of prescribed switching manifolds. When in thesliding mode, the system exhibits invariance properties, such as robustnessto certain internal parameter variations and external disturbances.

The dynamic performance of a VSS system is determined by the pre-scribed switching manifolds upon which the control structure is switched.Most commonly used switching manifolds are the linear hyperplanes, whichguarantee the asymptotic stability of the system motion in the sliding mode.That is, the system state will reach the equilibrium in infinite time. Thisasymptotical stability feature is, in general, sufficient for delivering effectivecontrol but there are cases which may not be dealt with properly withoutimposing substantial control efforts. For example, when high precision andstringent reaching time are required, controllers enabling asymptotical sta-bility may not perform well when the system state is close to the equilibrium.One alternative to overcome this problem is by means of specially tailorednonlinear switching manifolds. The recently developed finite time mecha-nism, the terminal sliding mode, is a useful nonlinear switching manifoldwhich can improve the transient performance substantially. The concept of


TSM was originated from the notion, terminal attractors [28], which wasused in studying the content addressable memory, associative memory, andpattern recognition in artificial neural networks. The TSM design was firstused in [12] for finite time sliding mode control design for robotic manipula-tors. It was then extended to several control problems of SISO systems andMIMO systems including robotics [24–26,13–15,4,23]. This chapter aims toexamine the current developments in this special topic and make some usefulextensions of existing results for control analysis and design using TSM.

2 The Terminal Sliding Mode Concept

The TSM concept can be described by the following first order dynamics [24]

s = x1 + β xq/p1 = 0 (1)

where x1 ∈ R1 is a scalar variable, β > 0, and p, q (p > q) are positiveintegers. Note that the parameter p must be an odd integer and only realsolution is considered so that for any real number x1, x

q/p1 is always a real

number. The equation (1) then becomes

x1 = −βxq/p1 (2)

Given an initial state x1(0) = 0 the dynamics (1) will reach x1 = 0 in finitetime. The time taken from the initial state x1(0) to 0, ts, is determined by

ts =p

β(p− q)|x1(0)|(p−q)/p (3)

It can also be proved that the equilibrium 0 is an attractor, i.e. when thestate x1 reaches zero, it will stay at zero forever. This can be demonstratedby taking a Lyapunov function v = 1

2x21. The time derivative of v along (1)

isv = x1x1 = −βx1x

q/p1 = −βx

(p+q)/p1

since (p + q) is even, then v is negative definite, so x1 = 0 is “terminally”stable (not necessarily asymptotically stable).

The reaching time ts which is determined by (3) depends on the parame-ters p, q, β, and the initial value x1(0). As x1(0) is fixed or in a known boundedregion, one can choose β such that ts is very small.

It is interesting to note that when p = q, then

s = x1 + βx1 = 0 β > 0 (4)

which becomes a linear dynamics. The introduction of the nonlinearity termx

q/p1 improves the convergence toward the equilibrium. The closer to the

110 X. Yu and Z. Man

equilibrium, the faster the convergence rate, resulting in finite time conver-gence. This can be demonstrated as follows. Consider the Jacobian aroundthe equilibrium x1 = 0, i.e.

J =∂x1

∂x1= − βq

pxp−q

q

1

For scalar x1, J is the eigenvalue of the first order approximation matrix J .We have

J → −∞ when x1 → 0

which indicates that at the equilibrium the “eigenvalue” tends to negativeinfinity, and of course, the system trajectory with such an “infinitely” negativeeigenvalue, will converge to the equilibrium with an “infinitely” large speedwhich results in finite time reachability.

Note that here the Lipschitz condition does not hold, i.e. |J | < ∞ doesnot hold and J is singular at x1 = 0. This situation introduces a violation tothe Lipschitz condition for the existence and uniqueness of solutions of dif-ferential equations. With the Lipschitz condition, a transient solution cannotintersect the corresponding constant solution, therefore the theoretical timeof approaching the equilibrium is always infinite. Violating the Lipschitz con-dition may give rise to the solutions which may reach a so called “terminaltrajectory” and solutions may intersect each other [28].

It is should be noted that there is a close relationship between first orderTSM (1) and time optimal control. It is well established that the time optimalcontrol for the double integrator system [16]

x1 = x2, ; x2 = u; |u| ≤ 1

can be described asu = sgn[Ξ(x)]

where x = (x1, x2)T and

Ξ(x) =

ξ(x) = x1 + 12x2|x2|; ξ(x) = 0

x2; ξ(x) = 0

Function ξ(x) = 0 forms the arc on which the system trajectory will reachzero. One can easily see that if we take very large odd numbers p and q suchthat q/p ≈ 1/2 and β = 2p/q, then the first order dynamics (1)

s(x) = 2p/qxq/p1 + x2 = 0

is equivalent to

x1 +12x

p/q2 ≈ x1 +

12x2|x2| = ξ(x) = 0.

111Variable Structure Systems with Terminal Sliding Modes

This means that the TSM model can approximate the time optimal arc forthe double integretor system with any accuracy regardless of the sign of x2

because it is smooth hence differentiable. It should be noted that this TSMmodel is also the optimal switching function for second order systems withpositive real eigenvalues in simple ratio λ1/λ2 = 2 [16].

One can clearly see that when x1 is far away from the equilibrium, theTSM model (1) does not prevail over its linear counterpart (setting p = q)since the term x

q/p1 tends to reduce the magnitude of convergence rate at a

distance from the equilibrium. One immediate solution is to introduce thefollowing so called fast TSM model,

s = x1 + α x1 + β xq/p1 = 0 (5)

where x1 ∈ R1, α, β > 0. By doing so, we have

x1 = −αx1 − βxq/p1 (6)

For properly chosen q, p, α, β, given an initial state x1(0) = 0 the dynamics(6) will reach x1 = 0 in finite time. The physical interpretation is: When x1

is far away from zero, the approximate dynamics become x1 = −αx1 whosefast convergence when far away from zero is well understood. When close tox1 = 0, the approximate dynamics become x1 = −βx

q/p1 which is a terminal

attractor [28].More precisely, we can solve the differential equation (6) analytically. The

exact time to reach zero, ts, is determined by

ts =p

α(p− q)ln

αx1(0)(p−q)/p + β

β(7)

and the equilibrium 0 is a terminal attractor.The fast convergence performance of the fast TSM in comparison with

the conventional linear sliding mode can be demonstrated by the followingexample (using Matlab). Consider α = 1 and β = 1 and initial conditionx1(0) = 1. First let assume p = 3 amd q = 1. From (7) one can easilyfind that the time to reach zero is ts = 1.03972077083992. We now comparethe above with the situation where p and q are set to 1. The simulationsuggests that at approximate ts = 1.03969999999990, for the case of p = 3and q = 1, x1(ts) = 0.00000009178540 and for the case of p = 1 and q = 1,x1(ts) = 0.12500519281775. It is evident that the convergence rate of thefast TSM is far better than its linear counter part. The obvious reason iswhen close to the equilibrium, the convergence rate of the linear sliding modeexponetially slows down while the convergence rate of the fast sliding modeaccelerates exponentially.

In the following sections, we will illustrate how to use the TSM conceptfor control design for SISO systems and MIMO systems respectively.


3 Fast TSM Control of SISO Systems

We consider the general nonlinear smooth single-input single-output (SISO)system

z = f(z) + g(z)uy = h(z) (8)

where z ∈ Rn, f and g are the smooth vector fields on Rn, h is the scalarsmooth field on Rn, and u ∈ R1.

Before we proceed, we introduce several notations. The Lie derivative ofh with respect to f is defined as the directional derivative Lfh = h whereh = (∂h/∂z) representing the gradient of h. Higher order Lie derivativescan be defined recursively as Li

fh = (Li−1f h)f for i = 1, 2, . . . with L0

fh = h.see [8,18].

Without loss of generality, we make the following assumption. However,the results to be obtained can easily be extended to those systems withrelative degree less than n.

Assumption 1 The SISO system (8) has relative degree n in a region Σ ∈Rn, i.e.

LgLifh = 0 0 ≤ i < n− 1 (9)

LgLn−1f h = 0 (10)

With Assumption 1, in Σ, the nonlinear system (8) can be transformedto the following normal form

xi = xi+1 i = 1, . . . , n− 1xn = a(x) + b(x)u (11)

where xi = Li−1f h(z) for i = 1, . . . , n, a(x) = Ln

fh(z) and b(x) = LgLn−1f h(z).

The transformation matrix is easily seen (see [8], p157)

x = Φ(z) = [h(z),Lfh(z), . . . ,Ln−1f h(z)]T .

The dynamics (11) can also be written as

x = A(x) +B(x)u (12)

where A(x) = [x2, . . . , xn−1, a(x)]T and B(x) = [0, . . . , 0, b(x)]T .For this class of system in ‘nested’ canonical form, the following hierar-

chical TSM structure can be used:

s1 = s0 + α0s0 + β0sq0/p00 (13)

s2 = s1 + α1s1 + β1sq1/p11 (14)

......

sn−1 = sn−2 + αn−2sn−2 + βn−2sqn−2/pn−2n−2 (15)


where s0 = x1, αi > 0, βi > 0, and qi, pi (i = 0, · · · , n− 2) are positive oddnumbers. The same analogue applies that once sn−1 = 0 is reached, sn−2 willreach zero in finite time, and sn−3, · · · , s0 will also reach zero. It is easy toestablish that the time to reach the equilibrium is

T =n∑

i=1

ti (16)

where tn is the time to reach the terminal sliding mode sn−1 = 0 and

ti =pi−1

αi−1(pi−1 − qi−1)ln

αi−1si−1(ti)(pi−1−qi−1)/pi−1 + βi−1

βi−1

for i = n− 1, · · · , 1 is the time from si(ti) = 0 to si(ti + ti−1) = 0.

Remark 1. Note that the procedure (13)–(15) actually defines a path for thestate x to converge to the equilibrium. Indeed, if sn−1 = 0 is considered asn − 1 dimensional flow in the state space, sn−2 = 0 can be considered as asubspace with dimension shrunk by unity. So s0 = 0 will be the result of then−1 dimensional dynamic space shrunk by n−1 times (dimensions). Lettingpi = qi (i = 1, · · · , n− 1), simple computation and transformation using theLaplace operator p lead to

sn−1 = (p+ αn−1 + βn−1) · · · (p+ α1 + β1)x.

Therefore sn−1 becomes a conventional linear hyperplane based sliding modewith real eigenvalues. We can also interpret the convergence towards x = 0as a shrinking process the same as that for the TSMs. However here, becauseof asymptotic convergence, the exact reduction of dimension is not possible.

The sliding mode control u can be designed so that

sn−1sn−1 ≤ −K|sn−1| K > 0

which is a sufficient condition. Therefore sn−1 = 0 can be reached in finitetime, and x = 0 can be reached in finite time as well. Following are someresults on the fast TSM control of SISO systems.

Theorem 1. For the system (8), if the control u is designed as

u = ueq + ud (17)

where

ueq = −b−1(x)(a(x) +n−2∑k=0

(αkLn−k−1A+Bu sk + βkLn−k−1

A+Bu sqk/pk

k )),

ud = −b−1(x)K sgn(sn−1) K > 0

where q and p (p > q) are positive odd integers defined above with K > 0being a constant, then the system state will reach the sliding mode sn−1 = 0in finite time.


Proof. To ensure the finite time reachability of the sliding mode sn−1 = 0,the condition sn−1 sn−1 < −K|sn−1| should be satisfied. Take the first orderderivative of sn−1, one has

sn−1 = sn−2 + αn−2sn−2 + βn−2LA+Busqn−2/pn−2n−2 (18)

since si = si−1 + αi−1si−1 + βi−1sqi−1/pi−1i−1 , for i = n− 1, n− 2, . . . , 1, and

the lth order derivative of si is

LlA+Busi = Ll+1

A+Busi−1 + αi−1LlA+Busi−1 + βi−1Ll

A+Bu(sqi−1/pi−1i−1 )

then it can be easily calculated that

sn−1 = LnA+Bus0 +

n−2∑k=0

βkLn−k−1A+Bu sk +

n−2∑k=0

βkLn−k−1A+Bu s

qk/pk

k

= zn +n−2∑k=0

αkLn−k−1A+Bu sk +

n−2∑k=0

βkLn−k−1A+Bu s

qk/pk

k (19)

Substituting the control (17) into (19) yields sn−1sn−1 = −K|sn−1| whichmeans that the sliding mode sn−1 = 0 will then be reached in finite time. Infact, following Section 2, from any initial state at t0 = 0, the time to reachzero is tn = |sn−1(0)|

K . Hence the proof of the theorem is completed.

The control (17) involves calculation of the terms Ln−k−1A+Bu s

qk+1/pk+1k , k =

0, . . . , n−2 that is lengthy and trivial. Here we present a qualitative result forthe calculation of these terms and also show that these terms are independentof the control u.

Theorem 2. For any k ∈ 0, 1, . . . , n− 2,Ln−k−1

A+Bu sqk/pk

k = fk(x) (20)

where fk is a continuous nonlinear function.

Proof. We prove this proposition using the mathematical induction ap-proach. Let k = 0, then apparently Ln−1

A+Busq0/p00 = Ln−1

A+Buxq0/p01 = f0(x). Let

k = 1, then Ln−2A+Bus

q1/p11 = Ln−2

A+Bu(x2 + α0x1 + β0xq0/p01 )q1/p1 which can be

apparently expressed as f1(x). Assume for k = k0, Ln−k0−1A+Bu s

qk0/pk0k0

= fk0(x).Let us examine the case of k = k0 + 1. Since

sk0 = fk0(sk0−1, sk0−1)= fk0(sk0−2, sk0−2, sk0−2)

=...

= fk0(s(k0)0 , s

(k0−1)0 , . . . , s0)


and s0 = x1, then s(l)k0

is a function of xk0+l+1, xk0+l, . . . , x1. Therefore

Ln−k0−2A+Bu s

qk0+1/pk0+1

k0+1

is a function of x.

The parameters qk, pk must be chosen carefully in order to avoid a singu-larity because there are terms in dn−k−1

dtn−k−1 sqk/pk

k which may contain nagativepowers so that when sk−1 → 0, u → ∞. This problem can be remedied bythe following theorem.

Theorem 3. If

qk/pk >n− k − 1

n− k

then when sk → 0 sequentially from k = n− 2 to k = 0, u is bounded.

Proof. From the rule for the nth derivative of a composite function [9], wehave that for function F (s),

dn

dtnF (s) =

∑ n!i1!i2! . . . il!

dmF

dsm

(s

1!

)i1 (s

2!

)i2 (s(3)

3!

)i3

· · ·(s(l)

l!

)il

(21)

over all solutions in non negative integers of the equation i1 + 2i2 + 3i3 +. . . lil = n and m = i1 + i2 + i3 + . . . + il. For simplicity, let r = qk/pk anddrop the index k. Since in the sliding mode sk+1 = 0,

sk+1 = sk + αksk + βksqk/pk

k = s+ αs+ βsr,

then s = O(sr) when s → 0, where O is a complexity function. We thereforehave dmF

dsm = O(sr−m) and s(d) = (s)(d−1) = O(sdr−(d−1)). So

dn

dtnF (s) =

∑O(sr−m)O(si1r)O(si2(2r−1)) · · · O(sil(lr−(l−1)))

=∑

O(sr−m)×O(sr(i1+2i2+3i3+...+lil)+(i1+i2+i3+...+il)−(i1+2i2+3i3+...+lil))

=∑

O(sr−m)O(snr+m−n) = O(y(n+1)r−n) (22)

Hence when s → 0, i.e. sk → 0, (n+1)r−n = (n+1)qk/pk−n > 0 will ensurethat (22) is bounded. Also from the above analysis, we have dn

dtn s = (s)(n−1) =O(sr)(n−1) = O(snr−n+1). Hence when s → 0, nr−n+1 = nqk/pk−n+1 > 0will ensure that dn

dtn s is bounded. With the above expressions in mind, thecontrol (17) can be rewritten as

u = −b−1(z)(a(z) +n−2∑k=0

(O(s(n−k−1)qk/pk−(n−k−1)+1k ) +

O(s(n−k)qk/pk−(n−k−1)k )) +K sgn(sn−1)) (23)


For the second term of (23) to be bounded while sk → 0, it is sufficient that(n − k)qk/pk − (n − k − 1) > 0 for sk → 0 sequentially from k = n − 2 tok = 0 so that the control u is bounded.

Another kind of singularity is that during the transient process towardsthe fast TSMs, the control may become singular if some si becomes zero. Forexample, for the second order SISO system

x1 = x2

x2 = a(x) + b(x)u (24)

from Theorem 1, the controller can be derived as

u = −b−1(x)(a(x) + α0x2 +β0q

px

(q−p)/p1 x2 +Ksgn(s)) (25)

from which one can see when initially x2 = 0 while x1 = 0, the control ubecomes singular.

We now analyze the the dynamic characteristics of the fast TSM in rela-tion to the singular problem.

In order to overcome the singularity problem, an effective means is toprohibit the trajectory from reaching the switching surface si = 0 (i =0, 1, ..., j − 1) before sj = 0 is reached (for some j). The idea is to find aregion in the state space such that any trajectory starting from this regionwill not incur the singularity problem and the set of switching manifoldssn−1 = 0, ..., s1 = 0, s0 = 0 are reached sequentially. For this purpose, wedefine a set as

Ω = x : s0 > 0 ∩ x : s1 > 0 ∩ ... ∩ x : sn−1 > 0. (26)

We shall prove in the following that the trajectory starting from Ω will re-sult in the switching manifolds sn−1 = 0, ..., s1 = 0, s0 = 0 being reachedsequentially and the singularity problem will not occur.

Lemma 1. The set Ω is unempty open set, that is Ω = φ.

Proof. From the definition of sj (j = 1, ..., n−1) in (13–15), we know s0 = x1.Select any nonzero initial values s0(0) = x1(0) and x2(0) such that

x2(0) + α0s0 + β0sq0/p00 > 0,

then s1(0) = x2(0) +α0s0 + β0sq0/p00 > 0. In general, for sj (j = 1, ..., n− 1),

from its definition in (13–15) we get

sj = sj−1 + αj−1sj−1 + βj−1sqj−1/pj−1j−1

=j∑

i=1

(∂sj−1

∂xixi+1) + αj−1sj−1 + βj−1s

qj−1/pj−1j−1

=j−1∑i=1

(∂sj−1


qj−1/pj−1j−1 +

∂sj−1

∂xjxj+1.


According to the recursive structure (13–15) and the system (12), it followsthat sj−1 is independent of xj+1, ..., xn and ∂sj−1

∂xj= 1. Hence,

sj =j−1∑i=1

(∂sj−1


qj−1/pj−1j−1 + xj+1.

This guarantees that when the xi(0) (i = 1, ..., j) are given, one can alwaysfind a xj+1(0) such that sj(0) > 0. Doing it in order, we can always findinitial values x1(0), x2(0), ..., xn(0) such that si(0) > 0 (i = 0, ..., n− 1). Theproof of Ω being an unempty open set is easily followed by the fact that theintersection of a set of open sets is also an open set since each set x : sj > 0is an open set.

Lemma 2. For the system (12) with control law (17), if the initial conditionx(0) ∈ Ω, then trajectory x(t) will first reach sn−1 = 0 and then sn−2 =0,...,s0 = 0 sequentially.

Proof. From the recursive TSM structure (13–15), it follows that

sj = sj+1 − αjsj − βjsqj/pj

j , j = 1, · · · , n (27)

Since x(0) satisfies sj > 0 (j = 1, ..., n− 1), from the dynamics (27) one cansee that if sn−2(t) reaches zero firstly while sn−1(t) > 0 we obtain at thetime that sn−2 = 0 but sn−2 = sn−1 > 0. This contradicts the fact thatsince initially sn−2 > 0, sn−1 > 0, then sn−2 > 0 which means sn−2(t) willnever decrease before sn−1(t) becomes zero. This implies that the trajectorystarting from x(0) ∈ Ω must reach the switching surface sn−1 = 0 first beforereaching sn−2 = 0. When sn−1 = 0, from (27), it follows that

sn−2sn−2 = −αn−2 − βn−2s(qn−2+pn−2)/pn−2n−2 < 0

due to pn−2+ qn−2 > 0 and αn−2, βn−2 > 0. This shows that, once the statetrajectory reaches sn−2 = 0, it will be confined to this manifold. The sameanalogy applies to the other cases and we conclude that si, i = n− 1, · · · , 1reach zero sequentially.

Theorem 4. For the system (12), if the recursive TSM structure is selectedas (13)–(15), the control law is (17), and Lemma 2 holds, then for x(0) ∈ Ω,the state x(t) will reach zero in finite time and the control u(t) is bounded.

Proof. Since the switching manifolds si = 0 (i = n − 1, ..., 1) are reachedsequentially, we obtain that in finite time s0(t) = x1(t) will approach zero.From the TSM structure x2(t), ..., xn(t) will all approach zero in finite time.Since

d

dt(sqi/pi

i ) =qi

pis(qi−pi)/pi

i si =qi

pis(2qi−pi)/pi

i , (28)


and the trajectory reaches si+1 = 0 earlier than si = 0, and Lemma 2 holds,then the term d

dt (sqi/pi

i ) is bounded. For high order derivatives, since theconditions in Lemma 2 hold, the same analogy applies. Therefore, we concludethat ueq(t) as well as u(t) is bounded for x(0) ∈ Ω.

Remark 2. The analysis shown above shows that the system trajectory willreach zero in finite time if x(0) ∈ Ω. However if an x(0) ∈ Ω, then thesingularity problem may occur. In this case the problem can be overcome bythe two phase control proposed in [20].

Remark 3. For the nonlinear system (8) with external disturbances and noisesv,

x = f(x) + g(x)u+ d(x)v, y = h(x),

if d(x) ∈ range(g(x)), i.e. the matching condition is satisfied [17], then a slightmodification of control (17) will guarantee the stability, the attainability ofthe sliding modes and finite time attainability of system equilibrium. Ourrecent study on robot control has shown that fast TSM control has bettercontrol precision and robustness than its linear counterpart. In fact, fast TSMcontrol is a high gain control when near the equilibrium.

Remark 4. The mechanism can easily be extended to the design of the outputtracking problem if z is replaced by e = zd−z, where zd is the desired outputsignals (for the problem statement, see [8,18]).

4 TSM Control of MIMO Systems

The TSM concept can be used for the control of MIMO systems [13]. Considerthe MIMO system represented as follows:

x1 = A11x1 +A12x2 (29)x2 = A21x1 +A22x2 +B2u (30)

where x1 = (x1,1, · · · , x1,n−m)T ∈ Rn−m, x2 = (x2,1, · · · , x2,m)T ∈ Rm aresystem states, A11, A12, A21, A22 are (n−m)×(n−m), (n−m)×m,m×(n−m)and m × m matrices respectively. The pair (A11, A12) is controllable, B2 isnonsingular and n ≤ 2m. The FTSM vector is chosen as

s = C1x1 + C2x2 + C3xq/p1 (31)

where s ∈ Rm, and C1, C2, C3 are constant m×(n−m), m×m, m×(n−m)matrices, respectively. xq/p

1 represents a equal sized vector whose entries arethe (q/p)th power of the entries of x1.


Theorem 5. For the MIMO system (29) and (30), if the control is designedas

u =

−(C2B2)−1 [(C1A11 + C2A21)x1

+ qpC3diag(x

(q−p)/p1 )(A11x1 +A21x2)

−(C1A12 + C2A22)x2 −K1(s/‖s‖)] for s = 0 & all x1,i = 0B−1

2 (−A21x1 −A22x2 −K2x2) for s = 0 & any x1,i = 0

(32)

with K1, K2 > 0 and i = 1, · · · , n − m, then s = 0 will be reached in finitetime.

Proof. Substituting the control (32) into the time derivative of the Lyapunovfunction V = (1/2)sT s yields V = −K1‖s‖ < 0 for s = 0 and all x1,i = 0(i = 1, · · · , n − m) which is the sufficient condition for the reachability ofs = 0. Details see [13]. Remark 5. It can be seen from the Theorem 5, at the points such that s = 0and any x1,i = 0, sT s may be singular (unbounded). The second part of thecontrol in (32) is to avoid these singularities. Indeed, when s = 0 and anyx1,i = 0, with the control, we have

x2 = −K2x2

with the solution x2(t) = M exp(−K2t). Therefore,

x1 = A11x1 +A12M exp(−K2t)

The second term on the right hand side of the equation plays the role ofdriving x1 away from zero.

Theorem 6. For the system (29) and (30) with the TSM control (32), if theTSM parameter matrices are designed as

A11 −A12C−12 C1 = 0 (33)

A12C−12 C3 = diag(ρ1, · · · , ρn−m) (34)

where ρi > 0, i = 1, · · · , n−m, then the system states will reach zero in finitetime in the TSM.

Proof. See [13]. When in the sliding mode, the condition 2q > p > q has to be satisfied in

order to avoid the singularity problem, similar to the SISO case [13].The condition (33) may be too strong. We now look at an alternative.

When the sliding mode s = 0 is reached, i.e.

x2 = −C−12 C1x1 − C−1

2 C3x1 (35)

Then substituting (35) into (29) leads to

x1 = (A11 −A12C−12 C1)x1 −A12C

−12 C3x

q/p1 (36)


With the conditions (33) and (34), (36) becomes

x1 = −diag(ρ1, · · · , ρn−m)x1 (37)

Since (37) is in diagonal form, it can be treated in the same way as for thescalar TSM.

However the conditions (33) can be relaxed. By a properly chosen C1 andC2, the matrix A11−A12C

−12 C1 can have all its eigenvalues on the right hand

side of the complex plane [29].Qualitatively, when the system state is far away from the equilibrium

x1 = 0, then in terms of magnitude of power, the dynamics is dominated by

x1 = (A11 −A12C−12 C1)x1 (38)

which is asymptotically stable. The state will reach a neighborhood of x1 = 0in finite time. When the system state is near the equilibrium x1 = 0, usingthe same reasoning as above, the dynamics is dominated by

x1 = −A12C−12 C3x

q/p1 (39)

which will give rise to a finite time attainability when near the equilibrium,because A12C

−12 C3 is diagonal and each entry of x1 reaches the zero in finite

time so x1 will reach the equilibrium in finite time.We will now look at the stability of the system at the equilibrium. For

convenience, denote L1 = A11 −A12C−12 C1, L2 = −A12C

−12 C3. Consider the

Lyapunov function candidate

V = xT1 x1

Differentiating it along the dynamics (38) leads to

V = xT1 x1 + xT

1 x1

= xT1 (L1 + LT

1 )x1 + (xq/p)TLT2 x1 + xT

1 L2xq/p1

= xT1 (L1 + LT

1 )x1 −n−m∑i=1

ρix(p+q)/p1,i (40)

where x1,i represents the ith entry of vector x1. If the following conditionholds,

Reλ(LT1 + L1) < 0 (41)

i.e. the real parts of the eigenvalues of LT1 + L1 are negative, since p+ q > 0

is even, then V < 0, which means the equilibrium is stable. Since we have C1

and C2 which have excessive entries to be used for tuning, condition (41) isnot hard to fulfil. In summary, if the following conditions are satisfied


1. The matrices C1 and C2 are chosen such that

Reλ(A11 −A12C−12 C1) < 0

andλ(A11 −A12C

−12 C1 + (A11 −A12C

−12 C1)T ) < 0

2. The matrices C2 and C3 are chosen such that

A12C−12 C3 = diag(ρ1, · · · , ρn−m)

for some ρi > 0, i = 1, · · · , n−m,

then the equilibrium is globally stable and will be reached in finite time.

5 Continuous Approximation of TSM Control

In general, when the TSM control strategies are implemented, due to theswitchings involved, chattering may occur which must be eliminated so thethe controller can perform properly. This can be achieved by smoothing outthe control discontinuity in a thin boundary layer neighboring the switchingmanifold. A common practice is to replace the switching function sgn by acontinuous saturation function

sat(s, φ) = s

φ |s(t)| < φ

sgn(s(t)) |s(t)| ≥ φ(42)

where φ is the width of the layer Sφ = x, |s(t)| < φ. It is clear thatoutside Sφ, the convergence toward s = 0 is maintained. Our interest is inthe dynamic behavior inside Sφ.

For the first order TSM (1), we have s(t) = x1+βxq/p1 subject to |s| < φ.

Define a Lyapunov function V = |x1|. Then its time derivative along thedynamics s(t) becomes

V = sgn(x1)x1

= sgn(x1)(s(t)− βxq/p1 )

= sgn(x1)s(t)− β|x1|q/p

< |s| − β|x1|q/p < φ− β|x1|q/p (43)

which indicates that x1 will converge to the region defined by

β|x1|q/p < φ

which leads to

|x1| <(

φ

β

)p/q

subject to |s| < φ (44)


The inequality (44) indicates a significant improvement in the steady stateerror compared with the linear switching manifold because q < p. If φ < β,then the power of p/q(> 1) would significantly reduce the steady state error.This is one of the advantages of using TSM (1). For the fast TSM (5) analyticexpression similar to (44) is not obvious but one can obtain an approximateinequality

|x1| <(

φ

α+ β

)p/q

for |x1| < 1.For the general SISO case, suppose the width of the boundary layer for

sn−1 is φ > 0. We can easily conclude that for the hierachical structure(13)–(15) with αi = 0 (i = 0, · · · , n− 2), we have

|sn−2(t)| <(

φ

βn−2

)pn−2/qn−2

subject to |sn−1| < φ

Then it is easily obtained that for s0(t) = x1, we have

|x1| < φ∏n−2

i=0

piqi

∏n−2i=0 β

∏i

j=0

pjqj

i

subject to |sn−1| < φ

6 Nonsingular TSM Control

One of the problems of using TSM control is its singularity. Although methodsto avoid the singularity problem have been discussed in the above sections,it would be ideal to have a finite time control mechanism that does not incurthe singularity problem. In this section, we discuss a simple nonsingular TSMcontrol.

The simple nonsingular TSM control is based on the following modifiedTSM model [4]

s = βx1 + xp/q2 (45)

where x = (x1, x2)T are the state of the second order system (24) and p, qare defined as before, and β > 0 is a design parameter. The key point of using(45) is that when differentiating s, it does not result in terms with negativepowers.

We now develop a controller for the second order system (24) to see howeffective the TSM model (45) is in terms of removing the singularity problem.

For the switching function (45), its time derivative along the dynamics(24) is

s = βx1 +p

qx

p/q−12 x2

= βx2 +p

qx

p/q−12 (f(x) + b(x)u) (46)


If the control u is taken as

u = −b−1(x)(f(x) + βq

px

(2q−p)/q2 +Ksgn(s))) (47)

then we havess = −K|x2|(p−q)/q|s| < 0

One question is whether the TSM s = 0 will be reached in finite time.The answer is yes. Indeed, substituting the control (47) into second equationof the second order system (24) yields

x2 = −q

px

(2q−p)/q2 −Ksgn(s) (48)

It can easily be seen that if x2 = 0, then (48) becomes

x2(t) = −Ksgn(s)

which suggests that x2 = 0 while x1 = 0 is not an attractor. For cases ofs > 0 and s < 0 , we obtain x2 < −K and x2 > K respectively. It meansthere exists a vicinity of x2 = 0, |x2(t)| < δ for a small δ > 0. Also we havethat x2 < −K for s > 0 and x2 > K for s < 0. Therefore the crossing of thetrajectory from one boundary of the vicinity x2 = δ to the other boundaryx2 = −δ for s > 0 and from x2 = −δ to x2 = δ for s < 0 is in finite time. Forthe region outside the |x2(t)| < δ, since the conventional sliding mode controlstructure is employed and s will not change sign before s = 0 is reached, thetime to reach the boundaries of the vicinity is finite. For the region outsidethe |x2(t)| < δ, the time to reach the boundaries of the vicinity is finite.Indeed, we can easily show that

ss < −δK|s|,which means the finite time reachability of the boundaries. Therefore we canconclude that the TSM s = 0 will be reached from anywhere in the statespace in finite time.

The nonsingular TSM model can be used for the control design of a classof nonlinear dynamical systems such as

x1 = f1(x1, x2) (49)x2 = f2(x1, x2) + g(x1, x2) +B(x1, x2)u (50)

where x1 ∈ Rn, x2 ∈ Rn, f1 and f2 are smooth vector functions and g rep-resents the uncertainties and disturbances satisfying ‖g(x1, x2)‖ < lg wherelg > 0, B is a nonsingular matrix and u ∈ Rn is the control vector. Such sys-tems can be found in mechnical systems such as the robot control systems.If we have that (x1, x2) = (0, 0) if and only if (x1, x1) = (0, 0), we can usethe following nonsingular TSM

s = Λx1 + xΓ1 (51)


where Λ = diag(β1, · · · , βn), (βi > 0), Γ = diag(γ1, · · · , γn) with γi = pi/qi

(1 < γi < 2) for i = 1, · · · , n, and xΓ1 is represented as

xΓ1 = (xγ1

11, · · · , xγn

1n)T

We also adopt the notion that

xΓ1 = Γdiag(xγ1−1

11 , · · · , xγn−11n )x1 (52)

which can be easily verified. Similar control as in (47) can be designed for(49) and (50). Consider the Lyapunov function

V =12sT s (53)

If the control u is chosen as

u = −(

∂f1

∂x2B(x1, x2)

)−1

(Ks

‖s‖ +∂f1

∂x2f2(x1, x2) +

∂f1

∂x1f1(x1, x2) +

Γ−1Λdiag(x2−β111 , · · · , x2−βn

1n )) (54)

then the time derivative of the Lyapunov function (53) becomes

V ≤ −Kmini(|x1,i|γi−1)‖s‖

which indicates the finite time convergence of s = 0.One can easily see that the nonsingular TSM control (54) does not involve

any terms which have negative powers. When in the sliding mode s = 0, wehave

βix1i + xγi

1i = 0

which is equivalent tox1i + β

1/γi

i x1/γi

1i = 0

whose finite time convergence is well documented in previous sections. Hencewe can claim the the nonsingular TSM control can deliver finite time conver-gence without any singularity.

7 Discussions and Conclusions

We have examined the dynamic properties of several TSM models and theirapplications in control design. Because the TSM does not satisfy the Lips-chitz condition, the usual sense of Lyapunov stability does not apply. The wellknown existence and uniqueness of solutions of differential equations does notapply either. The solutions may intersect each other. Although TSMs intro-duce certain degree of mathematical difficulty in analyzing their behavior,the introduction of a special form of nonlinearity does improve the systemsdynamical convergence. Future work can be pursued in applications of TSM


to control problems that require high precision and integration with othercontrol (or non control) methods such as optimal control. Preliminary stud-ies along this line have been conducted in [23], where the TSM is combinedwith the frequency shaping optimal sliding mode control to give rise to abetter dynamical performance, and in [27] where the TSM is used for deriva-tive signal estimators. There is a close relationship between the second ordersliding modes and higher order sliding modes [1,11] and the TSM which isworth pursuing as well.

References

1. Bartolini, G., Ferrara, A., Levant, A., and Usai, E. (1999). On second ordersliding mode controllers. Variable Structure Systems, Sliding Mode and Non-linear Control. K. D. Young and U. Ozguner (Eds.), Lecture Notes in Controland Information Sciences, Springer Verlag, 247, 329–350.

2. Bhat, S. P. and Berstein D. S. (1997). Finite time stability of homogeneoussystems. Proc. American Control Conference, 2513–1514.

3. Feng, Y., Han F., Yu, X., Stonier, D. and Man Z. (2000). Tracking precisionanalysis of terminal sliding mode control systems with saturation functions. Ad-vances in Variable Structure Systems: Analysis, Integration and Applications,Yu X. and Xu. J.-X. (eds), pp. 325–334, World Scientific, Singapore.

4. Feng, Y., Yu, X. and Man, Z. (2001). Non singular terminal sliding mode controland its applications to robot manipulators. Proceedings of 2001 IEEE Inter-national Symposium on Circuits and Systems, III, pp. 545-548, Sydney May2001.

5. Fuller, A. T. (1973). Proof of the time optimality of a predictive control strategyfor systems of higher order. International Journal of Control, 18(6), 1121–1127.

6. Gulko, F. B., Ya Kogan, B., Ya. Lerner, A., Mikhailov, N. N. and Bovoseltseva,Zh. A. (1964). A prediction method using high speed analog computers and itsapplications. Automation and Remote Control, 25(6), 803–813.

7. Haimo, V. T. (1986). Finite time controllers. SIAM Journal of Control andOptimization, 24. 760–770.

8. Isidori, A. (1989). Nonlinear Control Systems. Springer-Verlag, Berlin, Heidel-berg.

9. Jefferey, A. (1994). Table of Integrals, Series and Products. Academic Press,Inc., 5th Edition.

10. Lewis, F. L., Abdallah, C. T. and Dawson, D. M. (1993). Control of RobotManipulators. Macmillan Publishing.

11. Levant, A. (1993). Sliding order and sliding accuracy in sliidng mode control.International Journal of Control, 58, 1247–1263.

12. Man, Z., Paplinski, A. P. and Wu, H. R. (1994). A robust MIMO terminalsliding mode control for rigid robotic manipulators. IEEE Transactions on Au-tomat. Control, 39, 2464–2469.

13. Man, Z. and Yu, X. (1997). Terminal sliding mode Control of MIMO systems.IEEE Transactions on Circuits and Systems – Part I, 44, 1065–1070.

14. Man, Z., O’Day, M. and Yu, X. (1999). A robust adaptive terminal sliding modecontrol for rigid robotic systems. J. Intelligent & Robotic Systems, 24, 23–41.


15. Man, Z. and Yu, X. (1997). Adaptive terminal sliding mode tracking controlfor rigid robotic manipulators with uncertain dynamics. JSME InternationalJournal, 40(3), 493–502.

16. Ryan, E.P. (1982). Optimal Relay and Saturating Control System Synthesis,Peter Peregrinus Ltd. London.

17. Sira-Ramirez, H. (1989). Sliding regimes in general nonlinear systems: A rela-tive degree approach. International Journal of Control, 50 1487–1506.

18. Slotine , J.-J. E. and Li, W. (1991). Applied Nonlinear Control. Prentice-Hall,Englewood Cliffs, NJ.

19. Utkin, V.I. (1992). Sliding Modes in Control Optimization. Springer Verlag,Berlin, Heidelberg.

20. Wu, Y., Yu, X. and Man, Z. (1998). Terminal sliding mode control design foruncertain dynamic systems. Systems and Control Letters, 34(5), 281–288.

21. Xia, Y., Yu, X., Ledwich, G. and Oghanna, W. (2000). Self-tuning relay controldesign for MIMO systems with fast convergence. IEEE Trans. Circuits andSystems - Part I, 47(10), 1548–1552.

22. Xia, Y., Yu, X. and Oghanna, W. (2000). Adaptive robust fast control forinduction motors. IEEE Trans. Industrial Electronics, 47(4), 854–862.

23. Xu, J.-X. and Cao, W. J. (2000). Synthesized sliding mode control of a single-link flexible robot. International Journal of Control, 73(3), 197–209.

24. Yu, X. and Man, Z. (1996). Model reference adaptive control systems withterminal sliding modes. International Journal of Control, 64(6), 1165–1176.

25. Yu, X. and Man, Z. (1998). Multi-input uncertain linear systems with terminalsliding mode control. Automatica, 34(3), 389–392.

26. Yu, X. and Man, Z. (1996). On finite time convergence: Terminal sliding modes.Proceedings of 1996 International Workshop on Variable Structure Systems,164–168, Tokyo Japan.

27. Yu, X. and Xu, J.-X. (1996). A novel nonlinear signal derivative estimator.Electronics Letters, 31(16), 445–1447.

28. Zak, M. (1989). Terminal attractors in neural networks. Neural Networks, 2,259–274.

29. Zinober, A. S. I. (1993). Variable Structure and Lyapunov Control. Springer-Verlag, London.


Adaptive Backstepping Control

Ali Jafari Koshkouei, Russell E. Mills, and Alan S.I. Zinober

Department of Applied MathematicsThe University of SheffieldSheffield S10 2TNUK

Abstract. Adaptive backstepping algorithms for a class of nonlinear continuousuncertain processes with disturbances are considered. Sliding mode control usinga combined adaptive backstepping sliding mode control algorithm is also studied.The algorithms follow a systematic procedure for the design of adaptive controllaws for the output of observable minimum phase nonlinear systems. This class ofsystems may include unmatched uncertainty including disturbances and unmodelleddynamics. The design methods are based upon (i) the backstepping approach, and(ii) a combination of sliding and backstepping.

1 Introduction

The backstepping procedure is a systematic design technique for globallystable and asymptotically adaptive tracking controllers for a class of nonlinearsystems. The backstepping approach has been developed in the last decade[4]-[13] for the systematic design of controllers for nonlinear systems, bothwith and without unmatched parametric uncertainty. Various backsteppingcontrol design algorithms compiled in [8], provide a systematic framework forthe design of tracking and regulation strategies suitable for large classes ofnonlinear systems.Adaptive backstepping algorithms have been applied to systems which

can be transformed into a triangular form, in particular, the parametricpure feedback (PPF) form and the parametric strict feedback (PSF) form[4]. This method has been studied widely in recent years [4], [9], [15]-[17].When plants include uncertainty with lack of information about the boundsof unknown parameters, adaptive control is more convenient; whilst, if someinformation about the uncertainty, e.g. bounds, is available, robust control isusually employed. Sliding mode control (SMC) is a robust control method,and backstepping can be considered to be a method of adaptive control. Thecombination of these methods yields benefits from both approaches. A sys-tematic design procedure has been proposed to combine adaptive control andSMC for nonlinear systems with relative degree one [20].An algorithm for the synthesis of dynamical adaptive backstepping con-

trollers (Rıos-Bolıvar [11] and Rıos-Bolıvar et al [12]) yields a dynamical adap-tive backstepping algorithm overcoming the limitations of the static adaptivebackstepping algorithm designs of Krstic et al [9]. This algorithm allows one


to apply the backstepping approach to a larger class of uncertain nonlinearsystems, satisfying observability and minimum phase conditions, which maybe in either triangular or nontriangular canonical forms, i.e. not in PPF orPSF forms [15]-[17]. A symbolic algebra toolbox allows straightforward design[13] of dynamical backstepping control.The adaptive sliding backstepping control of parametric semi-strict feed-

back systems (PSSF) with disturbances has been studied by Koshkouei andZinober [5], [7]. The method ensures that the error state trajectories move ona sliding hyperplane. The sufficient condition for the existence of the slidingmode (in Rios-Bolıvar and Zinober [13],[14]) is not needed. The plant maycontain unmodelled terms and unmeasurable external disturbances, boundedby known functions. The classical backstepping method has been extendedto this class of systems [6] to achieve the output tracking of a dynamicalreference signal.A dynamic adaptive backstepping controller is presented in Section 2. It

is made more robust by including SMC (Section 3) and Second-Order SMC(Section 4). An improved adaptive backstepping method for PSSF systems isstudied in Section 5 while sliding backstepping is considered in Section 5.2.These approaches are extended to non-triangular systems in Sections 7 and8. Illustrative examples are given in Sections 6 and 9 and some conclusionsare presented in Section 10.

2 Dynamical Adaptive Backstepping

The Dynamical Adaptive Backstepping (DAB) algorithm was developed byRıos-Bolıvar [11], [12]. It combines the backstepping approach with tuningfunctions, developed by Krstic [9], with a dynamic input-output linearization.It works for nonlinear uncertain systems satisfying observability and mini-mum phase conditions, which may be in either triangular or nontriangularforms. It generates an adaptive nonlinear control for regulation or tracking inan iterative and systematic fashion. An interlaced tuning function parameterestimate update law compensates for uncertain parameters. Consider

x = f0(x) + Φ(x)θ +(g0(x) + Ψ(x)θ

)u (1)

y = h(x)

where x ∈ n is the state; u, y ∈ the input and output respectively; andθ = [θ1, . . . , θp]T is a vector of unknown parameters. f0, g0 and the columnsof the matrices Φ, Ψ ∈ n×p are smooth vector fields in a neighbourhood R0

of the origin x = 0 with f0(0) = 0, g0(0) = 0; and h is a smooth scalarfunction also defined in R0. It is assumed the relative degree of (1) withrespect to u is 1 ≤ ρ ≤ n.The steps leading to the the design of the dynamical adaptive compen-

sator, follow an input-output linearization procedure in which, at each step, a

130 A.J. Koshkouei, R.E. Mills, and A.S.I. Zinober

control dependent nonlinear mapping and a tuning function are constructed.The parameter update law and the dynamical adaptive control law whichstabilize the controlled plant, are designed at the final step. This method isapplicable to a class of nonlinear systems which satisfying observable andminimum phase conditions. Therefore the following conditions should be as-sumed.

Assumption 1 System (1) is locally observable in a subspace R1 ⊂ R0 ⊂n.

Assumption 2 System (1) is minimum phase in R1 ⊂ R0 ⊂ n.

A full proof is in [12]. The algorithm is as follows:

DAB Algorithm

Coordinate transformation

z1 = y − yr(t) = h(0)(x)− yr(t) (2)

zk = h(k−1)(·)− y(k−1)r (t) + αk−1(·), 2 ≤ k ≤ n

with

h(k) =∂h(k−1)

∂θτk +

∂h(k−1)

∂x

[f0 + Φθ +

(g0 + Ψθ

)v1

]

+k−ρ−1∑

i=1

∂h(k−1)

∂vivi+1 +

∂h(k−1)

∂t(3)

ωk =

(∂h(k−1)

∂x+∂αk−1

∂x

)(Φ(x) + v1Ψ(x)

)(4)

αk = zk−1 +

(k−1∑i=2

zi∂h(i−1)

∂θ+

k−1∑i=3

zi∂αi−1

∂θ

)ΓωT

k

+k−ρ−1∑

i=1

∂αk−1

∂vivi+1 +

∂αk−1

∂θτk +

∂αk−1

∂t

+∂αk−1

∂x

[f0 + Φθ +

(g0 + Ψθ

)v1

]+ ckzk (5)

τk = Γk∑

i=1

ωTk zk (6)

Parameter update law

˙θ = τn = ΓWT z = Γ

[ωT

1 ωT2 . . . ωT

n

]z (7)

131Adaptive Backstepping Control

Dynamical adaptive compensator

v1 = v2

v2 = v3

... (8)

vn−ρ =1(

∂h(n−1)

∂vn−ρ+ ∂αn−1

∂vn−ρ

)[− zn−1 + y(n)

r − ∂h(n−1)

∂t− ∂αn−1

∂t

−(∂h(n−1)

∂x+∂αn−1

∂x

)[f0 + Φθ +

(g0 + Ψθ

)v1

]

+

(n−1∑i=2

zi∂h(i−1)

∂θ+

n−1∑i=3

zi∂αi−1

∂θ

)ΓωT

n

−(∂h(n−1)

∂θ+∂αn−1

∂θ

)τn − cnzn

−n−ρ−1∑

i=1

(∂h(n−1)

∂vi+∂αn−1

∂vi

)vi+1

]

with v1 = u, the ci’s constant design parameters and Γ = ΓT > 0 theadaptation gain matrix. The control u is obtained implicitly as the solutionof the nonlinear time-varying differential equation (8).

3 Dynamical Adaptive Backstepping with Sliding

The robust combination of sliding mode and adaptive control methods hasbeen studied in recent years. To provide robustness, the adaptive backstep-ping algorithm can be modified to obtain adaptive sliding output trackingcontrollers. The modification is carried out at the final step of the algorithmby incorporating the sliding surface, defined in terms of the error coordinates

σ = k1z1 + . . .+ kn−1zn−1 + zn = 0 (9)

where the scalar coefficients ki > 0, i = 1, . . . , n − 1, are chosen in such amanner that the polynomial

p(s) = k1 + k2s+ . . .+ kn−1sn−2 + sn−1 (10)

in the complex variable s is Hurwitz. The update law is

˙θ = τn = τn−1 + Γσ

(ωn +

n−1∑i=1

kiωi

)

= Γ( n−1∑

i=1

ziωi + σ(ωn +

n−1∑i=1

kiωi

))(11)


and the dynamical adaptive sliding mode output tracking controller is foundvia

zn−1 + h(n)(·)− y(n)r (t) + αn(·)

+

(n−1∑i=2

zi∂h(i−1)

∂θ+

n−1∑i=3

zi∂αi−1

∂θ

)Γ

(ωn +

n−1∑i=1

kiωi

)

−n−1∑i=1

ki

i−1∑

j=2

zj∂h(j−1)

∂θ+

i−1∑j=3

zj∂αj−1

∂θ

Γωi

+n−1∑i=1

ki

(∂h(i−1)

∂θ+∂αi−1

∂θ

)(τn − τi)

+n−1∑i=1

ki(−zi−1 − cizi + zi+1)

= −κσ − β sign(σ) (12)

with κ > 0, β > 0 and αn defined by

αn(·) = ∂αn−1

∂θτn +

∂αn−1

∂x

[f0 + Φθ + (g0 + Ψθ)u

]

+n−ρ−1∑

i=1

∂αn−1

∂u(i−1)u(i) +

∂αn−1

∂t(13)

4 Dynamical Adaptive Backstepping withHigher-Order Sliding

The main characteristic of sliding mode control is the theoretically infiniteswitching of the control law, which provides robustness against measurementerrors and unmodelled dynamics. In practice, however, the switching occursat a finite rate. This can lead to the chattering effect, an unwanted highfrequency vibration of the controlled plant.Higher order sliding modes (HOSM) may be used to eliminate the chat-

tering and provide improved accuracy in the steady state [10]. Sliding modesσ ≡ 0 may be classified by the number r of the first total derivative σ(r)

which is not a continuous function of the state variables. This number r iscalled the sliding order. The r-th order sliding mode is determined by theequalities

σ = σ = σ = . . . = σ(r−1) = 0

The standard sliding mode is of the first order. The motion in the r-th ordersliding mode is the same as in an ideal 1st order sliding mode, while the


switching is moved to the higher derivatives of the sliding surface, removingthe chattering.The precision of the standard sliding mode is proportional to the time

interval between measurements τ , while r-th order sliding modes may provideup to the r-th order of sliding precision i.e. max |σ| ≤ Cτ r.We present the DAB procedure with a second-order sliding mode con-

troller (SOSMC) used at the final step. This improves the accuracy of theoriginal DAB algorithm and greatly simplifies the control. A previous DAB-SOSMC algorithm used a law devised by Bartolini [1],[18]. This new methodreplaces that law with one by Levant [10], known as the twisting algorithm.This generates a simpler control structure.

4.1 Twisting Algorithm

The twisting algorithm is

u =

−u |u| > 1−αm sign (σ) |u| ≤ 1, σσ ≤ 0−αM sign (σ) |u| ≤ 1, σσ > 0

(14)

with 0 < αm < αM . This has a finite time convergence to the manifoldσ = σ = 0 and is valid even with large disturbances. For calculation purposes,the first difference of σ, ∆σ = σ(ti+1) − σ(ti), can be used instead of thederivative. The algorithm can also be rearranged to

u =−u |u| > 1− 1

2 ((αm + αM ) sign (σ) + (αM − αm) sign (∆σ)) |u| ≤ 1 (15)

The region |u| < 1 is obviously attractive. If we define

Lu(·) = ∂

∂x(·)x+ ∂

∂t(·)

as the total derivative with respect to time, considering u as constant, then

σ =d

dtLu (σ)

= Lu (Lu (σ)) +∂

∂uLu(σ) u

Assuming that

|LuLuσ| ≤ C

0 < K1 <∂

∂uLu(σ) < K2

then we can say that

σ ∈ [−C,C] + [K1,K2] u


Thus the trajectory in the σ, σ plane is bounded by the majorant curves [2]

|σ|+ 12

σ2

K1αM − C= constant, σσ > 0

|σ|+ 12

σ2

K2αm + C= constant, σσ ≤ 0

and ‘twists’ into the origin inside these curves (Fig. 1).

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6−4

−3

−2

−1

0

1

2

3

4

5

σ

σ

σ

0

1

.

.

.

2

Fig. 1. Bounding curve for twisting algorithm

If the intersections with the σ = 0 axis are denoted by σi, (i = 0, 1, 2, . . .),then∣∣∣∣ σi+1

σi

∣∣∣∣ <√

K2αm + C

K1αM − C< 1 (16)

The time taken for the trajectory to move from |σi| to |σi+1| is

Ti ≤ |σi|(K1αM − C)

+|σi|

(K1αm − C)

≤ 2 |σi|(K1αm − C)

(17)

Thus the origin is reached, and in a finite time.

4.2 DAB-TWISTING algorithm

By linking the DAB process and a modified twisting algorithm, we can cre-ate a second order backstepping sliding mode controller. Stopping the DABprocess at the nth stage, we have the n error variables given by (2). We thengenerate the sliding surface (9) (an improved sliding function compared with[18]) and use the Lyapunov function

Vn =12

n−1∑i=1

z2i +

12σ2 +

12

(θ − θ

)T

Γ−1(θ − θ

)(18)


Taking the derivative along the manifold σ = 0 i.e. zn = −∑n−1i=1 kizi, we

have

Vn = −n−1∑i=1

ciz2i −

n−1∑i=1

kizizn−1 +(θ − θ

)T

Γ−1( ˙θ − τn−1

)

+

(n−1∑i=2

zi∂h(i−1)

∂θ+

n−1∑i=3

zi∂αi−1

∂θ

)( ˙θ − τn−1

)+ σ (σ + zn−1)

We can choose the dynamic control law

v1 = v2

v2 = v3

...vn−ρ = vn−ρ+1

vn−ρ+1 = −12

((αm + αM ) sign (σ) + (αM − αm) sign (∆σ)

)(19)

with v1 = u. With this, we can attain σ ≡ 0 in a finite time. From thedefinition of σ, this means that the error variables will tend to zero. Also, bytaking the parameter estimate update law ˙θ = τn−1, the Lyapunov functionhas derivative

Vn = −n−1∑i=1

ciz2i −

n−1∑i=1

kizizn−1

= − [z1 . . . zn−1]TQ [z1 . . . zn−1]

≤ 0where Q ∈ (n−1)×(n−1)

Q =

c1 0 . . . 00 c2 . . . 0....... . .

...k1 k2 . . . kn−1 + cn−1

> 0 (20)

Hence the error variables zi will tend to zero, and in particular h(x)− yr(t)→ 0.

5 Parametric Semi-Strict Feedback Systems

We next consider the parametric semi-strict feedback form (PSSF) [5]-[7],[19]with disturbances

xi = xi+1 + ϕTi (x1, x2, . . . , xi)θ + ηi(x,w, t), 1 ≤ i ≤ n− 1

xn = f(x) + g(x)u+ ϕTn (x)θ + ηn(x,w, t) (21)

y = x1


where x = [x1, x2, . . . , xn]T is the state, y the scalar output, u the scalarcontrol and ϕi(x1, . . . , xi) ∈ p, i = 1, . . . , n, are known functions which areassumed to be sufficiently smooth. θ ∈ p is the vector of constant unknownparameters, ηi(x,w, t), i = 1, . . . , n, are unknown nonlinear scalar functionsincluding all the disturbances, and w is an uncertain time-varying parameter.

Assumption The functions ηi(x,w, t), i = 1, . . . , n are bounded by knownpositive functions hi(x1, . . . xi) ∈ p, i.e.

|ηi(x,w, t)| ≤ hi(x1, . . . xi), i = 1, . . . , n. (22)

The output y should track a specified bounded reference signal yr(t) withbounded derivatives up to n-th order. Assume that θ = θ − θ where θ(t) isan estimate of the unknown parameter θ.

5.1 Backstepping Algorithm

The PSSF algorithm is summarized as follows [6]:

PSSF Algorithm

Coordinate transformation

z1 = x1 − yr

zk = xk − αk−1 − y(k−1)r (23)

with

ωk = ϕk(x1, . . . , xk)−k−1∑i=1

∂αk−1

∂xiϕi(x1, . . . , xi)

ζk =n

4εeat

(h2

k +k−1∑i=1

(∂αk−1

∂xi

)2

h2i

)(24)

ξk = ηk −k−1∑i=1

∂αk−1

∂xiηi

τk = Γ

k∑i=1

ωizi (25)

αk = −zk−1 − ckzk − ωTk θ +

k−1∑i=1

∂αk−1

∂xixi+1 +

∂αk−1

∂t

−ζkzk +∂αk−1

∂θτk +

(k−2∑i=1

zi+1∂αi

∂θ

)Γωk (ck > 0)


Parameter update law

˙θ = τn = Γ

n∑i=1

ωizi (26)

Control law

u =1

g(x)[ − zn−1 − cnzn − f(x)− ωT

n θ +n−1∑i=1

∂αn−1

∂xixi+1

+∂αn−1

∂θτn +

∂αn−1

∂t−

(n−2∑i=1

zi+1∂αi

∂θ

)Γwn +y(n)

r − ζnzn

](27)

with cn > 0.

5.2 Sliding Backstepping Control

The parameter update law is again (11) and the adaptive sliding mode outputtracking controller is found by

u =1

g(x)[ − zn−1 − f(x)− ωT

n θ +∂αn−1

∂θτn +

n−1∑i=1

∂αn−1

∂xixi+1

+y(n)r +

∂αn−1

∂t− k1

(−c1z1 + z2 − n

4εh2

1z1eat)

−n−1∑i=2

ki( − zi−1 − cizi + zi+1 − ζizi − ∂αi−1

∂θ(τn − τi)

+

(i−2∑l=1

zl+1∂αl

∂θ

)Γwi

)+

(n−2∑i=1

zi+1∂αi

∂θ

)Γ

(ωn +

n−1∑i=1

kiωi

)

−Wσ −(K +

n∑i=1

kiνi

)sgn(σ)

], (28)

where kn = 1, K > 0 and W ≥ 0 are arbitrary real numbers, and

νi = hi +i−1∑j=1

∣∣∣∣∂αk−1

∂xi

∣∣∣∣hj , 1 ≤ i ≤ n (29)

6 Illustrative Example

Consider the second order system in PSSF form

x1 = x2 + x1θ +Ax21 cos(Bx1x2)

x2 = u (30)


where A and B are considered unknown but it is known that |A| ≤ 2 and|B| ≤ 3. We have

h1 = 2x21 ω1 = x1

z1 = x1 − yr z2 = x2 + x1θ + c1z1 + 2εx

41z1e

at − yr

α1 = −x1θ − c1z1 − 2εx

41z1e

at ω2 = −∂α1∂x1

x1

τ2 = Γ (ω1z1 + ω2z2) ζ2 = 2ε e

at(x1

∂α1∂x1

)2

0 5 100.1

0.2

0.3

0.4

0.5

0.6

t

x1(t)

0 5 10−1.5

−1

−0.5

0

0.5

t

x2(t)

0 5 10−1

0

1

2

3

4

t

Parameter estimate

0 5 10−8

−6

−4

−2

0

2

t

Control action

Fig. 2. Regulator responses with nonlinear control (31) for PSSF system

Then the control law (27) becomes

u = −z1 − c2z2 − ωT2 θ +

∂α1

∂x1x2 +

∂α1

∂θτ2 +

∂α1

∂t+ y(2)

r − ζ2z2 (31)

Simulation results showing desirable transient responses are presented inFig. 2 with yr = 0.4, a = 0.1, ε = 5, Γ = 0.5, c1 = 12 and c2 = 0.1.Alternatively, we can design a sliding mode controller for the system. As-sume that the sliding surface is σ = k1z1+ z2 = 0 with k1 > 0. The adaptivesliding mode control law (28) is

u = (c1k1 − 1) z1 − k1z2 − ωT2 θ +

∂α1

∂x1x2 +

∂α1

∂θτ2 +

∂α1

∂t+ y(2)

r

+12εh2

1z1eat −Wσ −

(K + k1 + |∂α1

∂x1|)h1sgn(σ) (32)

where τ2 = Γ (z1ω1 + σ(ω2 + k1ω1)). Simulation results showing desirabletransient responses are shown in Fig. 3 with yr = 0.05 sin(2πt), k1 = 1,K = 5, W = 0, a = 0.2, ε = 1, Γ = 0.05, A = 2, B = 3 and c1 = c2 = 2.


0 5 10 15 20−0.05

0

0.05

0.1

0.15

0.2

t

x1(t)

0 5 10 15 20−0.5

0

0.5

1

t

x2(t)

0 5 10 15 200.1

0.105

0.11

0.115

0.12

0.125

t

Parameter estimate

Fig. 3. Tracking responses with sliding control (32) for PSSF system

7 Disturbed DAB

In this section we extend the DAB to affine nonlinear systems with unmod-elled or external disturbances. We use a dynamical input-output linearizationand assume that there is a well-defined relative degree ρ. For systems wherethe output is a linearizing function, ρ = n, and the new algorithm collapsesto the approach of Section 5.

7.1 System

Consider

x = f0 + φθ + (g0 + ψθ)u+ η (33)y = h(x)

where x ∈ n is the state; u, y ∈ the input and output respectively; andθ = [θ1, . . . , θp]T is a vector of unknown parameters. f0, g0 and the columnsof the matrices Φ, Ψ ∈ n×p are smooth vector fields in a neighbourhood R0

of the origin x = 0 with f0(0) = 0, g0(0) = 0; and h is a smooth scalarfunction also defined in R0. η(x,w, t) ∈ n are unknown nonlinear scalarfunctions including all the disturbances and unmodelled dynamics. w is anuncertain time-varying parameter. It is assumed the relative degree (33) withrespect to u is 1 ≤ ρ ≤ n, and with respect to θ is one.

Assumption 3 The functions η(x,w, t) are bounded by known positive func-tions q(x)

|ηi(x,w, t)| ≤ qi(x) 1 ≤ i ≤ n (34)


Like the unperturbed version, the method is applicable to systems which areobservable (1) and minimum-phase (2).

7.2 DAB-EXTENDED Algorithm

For systems of the form (33), the general problem is to track adaptively abounded desired reference signal yr(t), with smooth and bounded derivativesup to n-th order, in the presence of unknown parameters θ and disturbancesη.

Step 1

Define the output tracking error

z1 = y − yr = h(0) − yr (35)

with h(0) = h. Then

z1 =∂h

∂x[f + φθ + (g + ψθ)u+ η]− yr

=∂h

∂x

[f + φθ +

(g + ψθ

)u]+∂h

∂x(φ+ ψu)

(θ − θ

)+∂h

∂xη − yr

= h(1) + ω1θ + ξ1 − yr

where θ(t) is an estimate of the unknown parameters θ and

h(1)(x, u, θ) =∂h

∂x

[f + φθ +

(g + ψθ

)u]

ω1 =∂h

∂x(φ+ ψu)

θ = θ − θ

ξ1 =∂h

∂xη

Consider the Lyapunov function

V1 =12z21 +

12θTΓ−1θ

where Γ is a symmetric positive definite matrix. The derivative of V1 is

V1 = z1z1 + θTΓ−1(− ˙θ

)

= z1

[h(1) + ξ1 − yr

]+ θTΓ−1

(ΓωT

1 z1 − ˙θ)

Set the first tuning function τ1 = ΓωT1 z1. If we can set −α1 = h(1) − yr,

where

α1(x, t) = (c1 + ζ1) z1

ζ1 =n

4εeat

n∑j=1

(∂h

∂xj

)2

q2j


where a, ε > 0, then we will have V1 ≤ 0. However, as this result is not validfor all time, we define the second error variable

z2 = h(1) + α1 − yr

giving the closed loop derivatives

z1 = −c1z1 + z2 + ω1θ + ξ1 − ζ1z1

V1 = −c1z21 + z1z2 + ξ1z1 − ζ1z

21 + θTΓ−1

(τ1 − ˙θ

)

Proceeding iteratively, we obtain the k-th step.

Step k, 1 ≤ k ≤ n− 1We have

zk = h(k−1) + αk−1 − y(k−1)r (36)

where

ωk =

(∂h(k−1)

∂x+∂αk−1

∂x

)(φ+ ψ v1) (37)

τk = Γk∑

j=1

ωTj zj (38)

h(k) =∂h(k−1)

∂x

[f + φθ +

(g + ψθ

)v1

]+∂h(k−1)

∂θτk +

∂h(k−1)

∂t

+k−ρ∑j=1

∂h(k−1)

∂vjvj+1 +

k−1∑j=2

zj∂h(j−1)

∂θΓωT

k (39)

ξk =

(∂h(k−1)

∂x+∂αk−1

∂x

)η (40)

ζk =n

4εeat

k∑j=1

(∂h(k−1)

∂xj+∂αk−1

∂xj

)2

q2j (41)

αk = zk−1 + ckzk +∂αk−1

∂x

[f + φθ +

(g + ψθ

)v1

]+∂αk−1

∂θτk

+∂αk−1

∂t+

k−ρ∑i=1

∂αk−1

∂vjvj+1 +

k−1∑j=3

zj∂αj−1

∂θΓωT

k + ζkzk (42)

The time derivative is

zk =∂h(k−1)

∂x[f + φθ + (g + ψθ) v1 + η] +

∂h(k−1)

∂θ

˙θ +

∂h(k−1)

∂t

+∂αk−1

∂x[f + φθ + (g + ψθ) v1 + η] +

∂αk−1

∂θ

˙θ +

∂αk−1

∂t


+k−ρ∑j=1

∂h(k−1)

∂vjvj+1 +

k−ρ∑j=1

∂αk−1

∂vjvj+1 − y(k)

r

=∂h(k−1)

∂x

[f + φθ +

(g + ψθ

)v1

]+∂h(k−1)

∂t+∂αk−1

∂t− y(k)

r

+∂αk−1

∂x

[f + φθ +

(g + ψθ

)v1

]+

k−ρ∑j=1

∂h(k−1)

∂vjvj+1

+k−ρ∑j=1

∂αk−1

∂vjvj+1 +

(∂h(k−1)

∂x+∂αk−1

∂x

)(φ+ ψ v1) θ

+∂h(k−1)

∂θ

˙θ +

∂αk−1

∂θ

˙θ +

(∂h(k−1)

∂x+∂αk−1

∂x

)η

= h(k) +∂αk−1

∂x

[f + φθ +

(g + ψθ

)v1

]+∂αk−1

∂θ

˙θ +

∂αk−1

∂t

+k−ρ∑j=1

∂αk−1

∂vjvj+1 − y(k)

r +∂h(k−1)

∂θ

( ˙θ − τk

)+ ωk θ

+ξk −k−1∑j=2

zj∂h(j−1)

∂θΓωk

where v1 = u, v2 = u, . . . , vj = u(j−1).Augmenting the Lyapunov function

Vk = Vk−1 +12z2k

=12

k∑j=1

z2j +

12θTΓ−1θ

⇒ Vk = Vk−1 + zkzk

= −k−1∑j=1

cjz2j +

k−1∑

j=2

zj∂h(j−1)

∂θ+

k−1∑j=3

zj∂αj−1

∂θ

( ˙

θ − τk−1

)

+k−1∑j=1

(ξjzj − ζjz

2j

)+ zk(zk−1 + h(k)

+∂αk−1

∂x

[f + φθ +

(g + ψθ

)v1

]+

k−ρ∑j=1

∂αk−1

∂vjvj+1 − y(k)

r

+∂h(k−1)

∂θ

( ˙θ − τk

)+ ωk θ +

∂αk−1

∂θ

˙θ +

∂αk−1

∂t


−k−1∑j=2

zj∂h(j−1)

∂θΓωT

k + ξk)+ θTΓ−1(τk−1 − ˙θ

)

If we can set −αk = h(k) − y(k)r , we will have Vk ≤ 0. However, as this

result is not valid for all time, we define the new error variable

zk+1 = h(k) + αk − y(k)r (43)

This gives

zk = −zk−1 − ckzk + zk+1 +

(∂h(k−1)

∂θ+∂αk−1

∂θ

)( ˙θ − τk

)

+ωk θ −k−1∑

j=2

zj∂h(j−1)

∂θ+

k−1∑j=3

zj∂αj−1

∂θ

ΓωT

k + ξk − ζkzk

Vk = −k−1∑j=1

cjz2j +

k−1∑

j=2

zj∂h(j−1)

∂θ+

k−1∑j=3

zj∂αj−1

∂θ

( ˙

θ − τk−1

)

+k−1∑j=1

(ξjzj − ζjz

2j

)+ θTΓ−1

(τk−1 − ˙θ

)+ zk (−ckzk + zk+1

+ωkθ + ξk − ζkzk +

(∂h(k)

∂θ+∂αk

∂θ

)( ˙θ − τk

)

−k−1∑

j=2

zj∂h(j−1)

∂θ+

k−1∑j=3

zj∂αj−1

∂θ

ΓωT

k

= −k∑

j=1

cjz2j +

k∑

j=2

zj∂h(j−1)

∂θ+

k∑j=3

zj∂αj−1

∂θ

( ˙

θ − τk

)

+k∑

j=1

(ξjzj − ζjz

2j

)+ zkzk+1 + θTΓ−1

(τk − ˙θ

)

Step n

At this step, the actual update law ˙θ = τn and the dynamical controller areobtained.We have zn = h(n−1) + αn−1 − y

(n−1)r , so

zn =∂h(n−1)

∂x[f + φθ + (g + ψθ) v1 + η] +

∂h(n−1)

∂θ

˙θ +

∂h(n−1)

∂t

+∂αn−1

∂x[f + φθ + (g + ψθ) v1 + η] +

∂αn−1

∂θ

˙θ +

∂αn−1

∂t


+n−ρ∑j=1

∂h(n−1)

∂vjvj+1 +

n−ρ∑j=1

∂αn−1

∂vjvj+1 − y(n)

r

=

(∂h(n−1)

∂x+∂αn−1

∂x

)[f + φθ +

(g + ψθ

)v1

]− y(n)

r + ωnθ

+n−ρ∑j=1

(∂h(n−1)

∂vj+∂αn−1

∂vj

)vj+1 +

∂h(n−1)

∂t+∂αn−1

∂t

+

(∂h(n−1)

∂θ+∂αn−1

∂θ

)˙θ + ξn

with

ωn =

(∂h(n−1)

∂x+∂αn−1

∂x

)(φ+ ψ v1)

ξn =

(∂h(n−1)

∂x+∂αn−1

∂x

)η

Augmenting the Lyapunov function

Vn = Vn−1 +12z2n

=12

n∑j=1

z2j +

12θTΓ−1θ

⇒ Vn = Vn−1 + znzn

= −n−1∑j=1

cjz2j +

n−1∑

j=2

zj∂h(j−1)

∂θ+

n−1∑j=3

zj∂αj−1

∂θ

( ˙

θ − τn−1

)

+n−1∑j=1

(ξjzj − ζjz

2j

)+ zn

(zn−1 +

∂h(n−1)

∂t+∂αn−1

∂t

+

(∂h(n−1)

∂x+∂αn−1

∂x

)[f + φθ +

(g + ψθ

)v1

]− y(n)

r

+n−ρ∑j=1

(∂h(n−1)

∂vj+∂αn−1

∂vj

)vj+1 + ωn θ

+

(∂h(n−1)

∂θ+∂αn−1

∂θ

)˙θ + ξn

)+ θTΓ−1

(τn−1 − ˙θ

)

We would like to have(∂h(n−1)

∂x+

∂αn−1

∂x

)[f + φθ +

(g + ψθ

)v1

]+∂h(n−1)

∂t+∂αn−1

∂t


−y(n)r + zn−1 +

n−ρ−1∑j=1

(∂h(n−1)

∂vj+∂αn−1

∂vj

)vj+1

= −cnzn − ζnzn −(∂h(n−1)

∂θ+∂αn−1

∂θ

)τn

−n−1∑

j=2

zj∂h(j−1)

∂θ+

n−1∑j=3

zj∂αj−1

∂θ

ΓωT

n

Thus our control law u = v1 can be found from

v1 = v2

v2 = v3

... (44)

vn−ρ =−1

∂h(n−1)

∂vn−ρ+∂αn−1

∂vn−ρ

[(∂h(n−1)

∂x+∂αn−1

∂x

)[f + φθ +

(g + ψθ

)v1

]

+∂h(n−1)

∂t+∂αn−1

∂t+

n−1∑

j=2

zj∂h(j−1)

∂θ+

n−1∑j=3

zj∂αj−1

∂θ

ΓωT

n

+

(∂h(n−1)

∂θ+∂αn−1

∂θ

)τn + zn−1 + cnzn − y(n)

r

+n−ρ−2∑

j=1

(∂h(n−1)

∂vj+∂αn−1

∂vj

)vj+1 + ζnzn

and choosing the parameter estimate update law to be

˙θ = τn = Γ

n∑j=1

ωj zj

we get

Vn = −n∑

j=1

cjz2j + θTΓ−1

(τn − ˙θ

)+

n∑j=1

(ξjzj − ζjz

2j

)

+

n∑

j=2

zj∂h(j−1)

∂θ+

n∑j=3

zj∂αj−1

∂θ

( ˙

θ − τn

)

≤ −n∑

j=1

cjz2j + nε e−at

Hence, by Barbalat’s Lemma, the system is stable.


8 DAB-EXTENDED-SMC

Robustness can be added to the disturbed DAB algorithm by having slidingmode control at the final stage. This disturbed DAB-SMC algorithm gener-ates the error variables in the same way as the disturbed DAB algorithm. Atthe n-th step, use the sliding surface (9) and the Lyapunov function

Vn = Vn−1 +12σ2

=12

n−1∑i=1

z2i +

12σ2 +

12θTΓ−1θ

⇒ Vn = Vn−1 + σσ

= −n−1∑i=1

ciz2i +

n−1∑

j=2

zj∂h(j−1)

∂θ+

n−1∑j=3

zj∂αj−1

∂θ

( ˙

θ − τn−1

)

+n−1∑j=1

(ξjzj − ζjz

2j

)+ zn−1zn + σ

[∂h(n−1)

∂t+∂αn−1

∂t

+

(∂h(n−1)

∂x+∂αn−1

∂x

)[f + φθ +

(g + ψθ

)v1

]

+n−ρ−1∑

j=1

(∂h(n−1)

∂vj+∂αn−1

∂vj

)vj+1 − y(n)

r + ωnθ

+

(∂h(n−1)

∂θ+∂αn−1

∂θ

)˙θ + ξn +

n−1∑i=1

ki (−zi−1 − cizi

+zi+1 + ωiθ +

(∂h(i−1)

∂θ+∂αi−1

∂θ

)( ˙θ − τi

)ξi − ζizi

)

−i−1∑

j=2

zj∂h(j−1)

∂θ+

i−1∑j=3

zj∂αj−1

∂θ

ΓωT

i

+ θTΓ−1(τn−1 − ˙θ

)

Setting ˙θ = τn = τn−1 + Γ(ωT

n +∑n−1

i=1 kiωTi

)σ, and since zn = σ −∑n−1

i=1 kizi we can rearrange to give

Vn = −n−1∑i=1

ciz2i − zn−1

n−1∑i=1

kizi +n−1∑j=1

(ξjzj − ζjz

2j

)

+σ

[zn−1 +

(∂h(n−1)

∂x+∂αn−1

∂x

)[f + φθ +

(g + ψθ

)v1

]


+∂h(n−1)

∂t+∂αn−1

∂t+

n−ρ−1∑j=1

(∂h(n−1)

∂vj+∂αn−1

∂vj

)vj+1

+

(∂h(n−1)

∂θ+∂αn−1

∂θ

)τn − y(n)

r + ξn

+

n−1∑

j=2

zj∂h(j−1)

∂θ+

n−1∑j=3

zj∂αj−1

∂θ

Γ

(ωT

n +n−1∑i=1

kiωTi

)

+n−1∑i=1

ki

(−zi−1 − cizi + zi+1 +

(∂h(i−1)

∂θ+∂αi−1

∂θ

)(τn − τi)

+ξi − ζizi −i−1∑

j=2

zj∂h(j−1)

∂θ+

i−1∑j=3

zj∂αj−1

∂θ

ΓωT

i

The dynamic controller can be found by solving

zn−1 +

(∂h(n−1)

∂x+∂αn−1

∂x

)[f + φθ +

(g + ψθ

)v1

]− y(n)

r

+

n−1∑

j=2

zj∂h(j−1)

∂θ+

n−1∑j=3

zj∂αj−1

∂θ

Γ

(ωT

n +n−1∑i=1

kiωTi

)

+

(∂h(n−1)

∂θ+∂αn−1

∂θ

)τn +

n−ρ−1∑j=1

(∂h(n−1)

∂vj+∂αn−1

∂vj

)vj+1

+n−1∑i=1

ki

(−zi−1 − cizi + zi+1 +

(∂h(i−1)

∂θ+∂αi−1

∂θ

)(τn − τi)

− ζizi −i−1∑

j=2

zj∂h(j−1)

∂θ+

i−1∑j=3

zj∂αj−1

∂θ

ΓωT

i

+ ∂h(n−1)

∂t+∂αn−1

∂t

= −λσ − β sgnσ − ζnσ − sgn(σ)n−1∑i=1

kiνi (45)

where

νi =i∑

j=1

∣∣∣∣∣∂h(i−1)

∂x+∂αi−1

∂x

∣∣∣∣∣ qj

This gives

Vn = −n−1∑i=1

ciz2i − zn−1

n−1∑i=1

kizi +n−1∑j=1

(ξjzj − ζjz

2j

)


+σ

[ξn − ζnσ +

n−1∑i=1

ki (ξi − νisgnσ)− λσ + β sgnσ

]

≤ − [z1 . . . zn−1]TQ [z1 . . . zn−1]− λσ2 − β |σ|+ nε e−at

≤ 0with Q from (20). This guarantees asymptotic stability for suitably chosendesign parameters.

9 Example: Robot with Flexible Joint

Sliding mode backstepping adaptive control offers the potential for controlof robotic manipulators in the presence of uncertain flexibilities, changingdynamics due to unknown and varying payloads, and nonlinear interactionwithout explicit parameter identification. In this section we consider a singlelink robot arm to illustrate the procedure of PSSF design (23)-(27).The system of a single link robot arm with a revolute elastic joint rotating

in a vertical plane is

v1 = v2

v2 =−Fl

Jlv2 − glM

Jlsin(v1) +

−kJl(v1 − v3) + η1

v3 = v4

v4 = −Fm

Jmv4 +

k

Jm(v1 − v3) +

1Jm

u+ η3

in which v1, v2, v3, and v4 are the link displacement, the velocity of the link,the rotor displacement and the velocity of the rotor, respectively. Jl is the linkinertia, Jm the motor rotor inertia, k the elastic constant,M the link mass, gthe gravity constant, l the centre of mass. The positive constant parametersFl and Fm are viscous friction coefficients. The bounded functions

η1 = η1(t, v1, v2, v3, v4, w)

andη2 = η2(t, v1, v2, v3, v4, w)

are the perturbation in the system including the motor disturbance. Thecontrol u is the torque delivered by the motor. It is assumed that the constantsM , Fl and Fm are unknown, and w is an uncertain time-varying parameter.The transformation

x1 = v1

x2 = v2

x3 =−kJl(v1 − v3)

x4 =−kJl(v2 − v4)


converts the system to

x1 = x2

x2 = x3 + ϕT2 θ + η2(t, x1, x2, x3, x4, w)

x3 = x4

x4 = −(k

Jl+

k

Jm

)x3 + ϕT

4 θ +k

JmJlu+ η4(t, x1, x2, x3, x4, w)

with

ϕT2 =

[−gl

Jlsin(x1) − 1

Jlx2 0

]

ϕT4 =

[glk

J2l

sin(x1)k

J2l

x2 − 1Jm

(k

Jlx2 + x4

)]

θ = [M Fl Fm]T

Suppose |η2| ≤ q2 and |η4| ≤ q4 where q2 and q4 are known constant realnumbers. It is desired for the output y = x1 to track the reference yr. Definethe sliding surface as

σ = k1z1 + k2z2 + k3z3 + z4 = 0

The error variables are

z1 = x1 − yr

z2 = x2 − α1(x1, t)

z3 = x3 − α2(x1, x2, θ, t)

z4 = x4 − α3(x1, x2, x3, θ, t)

with the stabilizing functions

α1 = −c1z1α2 = −z1 −

(c2 +

1εeatq2

2

)z2 − ϕT

2 θ − c1x2

α3 = −z2 −(c3 +

1εeat

(∂α2

∂x2

)2

q22

)z3 + ϕT

2

∂α2

∂x2θ +

∂α2

∂x1x2 +

∂α2

∂x2x3

+∂α2

∂θτ3 +

∂α2

∂t

The associated tuning functions are

τ2 = Γω2z2 = ΓϕT2 z2

τ3 = Γ (ω2z2 + ω3z3) = Γ

(z2 − ∂α2

∂x2z3

)ϕT

2

τ4 = τ3 + Γσ(ω4 + k3ω3 + k2ω2)

= Γ

[(z2 − ∂α2

∂x2z3

)ϕT

2 + σ

(ϕ4 +

(−∂α3

∂x2+ k2 − k3

∂α2

∂x2

)ϕT

2

)]


where Γ is an adaptation gain matrix. The update parameter law is ˙θ = τ4and the control is

u =JlJm

k[ − (c1k1 + k2)z1 + (k1 − c2k2 − k3)z2 + (k2 − c3K3 − 1) z3

+k3z4 +(k

Jl+

k

Jm

)x3 −

(ϕ4 − ∂α3

∂x2ϕ2

)T

θ −i=3∑i=1

∂α3

∂xixi+1

+∂α3

∂θτ4 +

∂α3

∂t−

(λ+

1εeat

(q24 +

(∂α3

∂x2

)2

q22

))σ

−(β + k2q2 + k3

∣∣∣∣∂α2

∂x2

∣∣∣∣ q2)sgn(σ)

−k3∂α2

∂θ(τ4 − τ3) +

∂α2

∂θΓ

(ϕ4 +

(−∂α3

∂x2+ k2 − k3

∂α2

∂x2

)ϕT

2

)z3

]

The simulation results are shown in Fig. 4 with the values Jl = 5 Nm2,Jm = 7 Nm2, l = 1 m, k = 400 Nm/rad. We assume that Fm = 0, Fl = 0h2 = 0.5, h4 = 0.1, η2 = 0.5 sin(x2

1), η4 = 0.1 cos(3x1x3) and yr = 3 asthe desired point. The design parameters were selected to be c1 = 2, c2 =0.1, c3 = 0.1, k1 = 2, k2 = 6, k3 = 1, λ = 5, β = 100, ε = 0.1, a = 3,Γ = 0.0095. The nominal unknown parameter M is taken to be 2 kg.

0 2 4 6 8 100

1

2

3

4

Time (sec)

Link

dis

plac

emen

t

0 2 4 6 8 10−4

−2

0

2

Time (sec)

Rot

or d

ispl

acem

ent

0 2 4 6 8 100

0.5

1

1.5

2

Time (sec)

Par

amet

er e

stim

ate

0 2 4 6 8 10−100

−50

0

50

Time (sec)

Con

trol

law

0 2 4 6 8 10−10

−5

0

5

10

Time (sec)

Slid

ing

func

tion

Fig. 4. Responses of Flexible Robot


10 Conclusions

Sliding mode control is a robust control method design and adaptive back-stepping is an adaptive control design method. In this chapter the controldesign has benefited from both design approaches. Backstepping control andsliding backstepping control were developed for a class of nonlinear systems.The plant may have unmodelled or external disturbances. The discontinuouscontrol may contain a gain parameter for the designer to select the velocity ofthe convergence of the state trajectories to the sliding hyperplane. The pre-vious work of Rios-Bolıvar and Zinober [13]-[14] and Koshkouei and Zinober[7], has been extended to include Second-Order SMC and to remove the suf-ficient existence condition to guarantee that the state trajectories convergeto the given sliding surface.

References

1. Bartolini G., Ferrara A., Usai E. (1998) Chattering Avoidance by Second-OrderSliding Mode Control. IEEE Trans. Aut. Cont. 43, 241–246

2. Emel’yanov S. V., Korovin S. K., Levant A (1993) Higher-Order Sliding Modesin Control Systems. Differential Equations. 29, 1627–1647

3. Freeman R. A., Kokotovic P. V. (1996) Tracking controllers for systems linearin unmeasured states. Automatica. 32, 735–746

4. Kanellakopoulos, I., Kokotovic P. V., Morse A. S. (1991) Systematic design ofadaptive controllers for feedback linearizable systems. IEEE Trans. Automat.Control. 36, 1241–1253

5. Koshkouei A. J., Zinober A. S. I. (2000) Adaptive Output Tracking Backstep-ping Sliding Mode Control of Nonlinear Systems. In: Proceedings of the 3rdIFAC Symposium on Robust Control Design, Prague, Czech Republic

6. Koshkouei A. J., Zinober A. S. I. (2000) Adaptive Backstepping Control ofNonlinear Systems with Unmatched Uncertainty. In: Proceedings of the CDC2000, Sydney, Australia

7. Koshkouei A. J., Zinober A. S. I. (1999) Adaptive Sliding Backstepping Controlof Nonlinear Semi-Strict Feedback Form Systems. In: Proceedings of the 7thIEEE Mediterranean Control Conference, Haifa, Israel

8. Krstic M., Kanellakopoulos I., Kokotovic P. V. (1995) Nonlinear and AdaptiveControl Design. John Wiley & Sons, New York

9. Krstic, M., Kanellakopoulos I., Kokotovic P. V. (1992) Adaptive nonlinear con-trol without overparametrization. Syst. & Control Letters. 19, 177–185

10. Levant A. (1993) Sliding order and sliding accuracy in sliding mode control.Int. J. Control. 58, 1247–1263

11. M. Rios-Bolıvar (1997) Adaptive Backstepping and Sliding Mode Control ofUncertain Nonlinear Systems. PhD dissertation, University of Sheffield.

12. Rios-Bolıvar M., Sira-Ramırez H., Zinober A. S. I. (1994) Output TrackingControl via Adaptive Input-Output Linearization: A Backstepping Approach.In: Proc. 34th CDC, New Orleans, USA. 2, 1579–1584

13. Rios-Bolıvar M., Zinober A. S. I. (1998) A symbolic computation toolbox forthe design of dynamical adaptive nonlinear control. Appl. Math. and Comp.Sci. 8, 73–88


14. Rios-Bolıvar M., Zinober A. S. I. (1997) Dynamical adaptive backstepping con-trol design via symbolic computation. In: Proceedings of the 3rd EuropeanControl Conference, Brussels, Belgium

15. Rios-Bolıvar M., Zinober A. S. I. (1997) Dynamical adaptive sliding mode out-put tracking control of a class of nonlinear systems. Int. J. Robust and NonlinearControl. 7, 387–405

16. Rios-Bolıvar M., Zinober A. S. I. (1996) Dynamical Sliding Mode Control viaAdaptive Input-Output Linearization: A Backstepping Approach. In: Garo-falo F., Glielmo L. (Eds.) Robust Control via Variable Structure and LyapunovTechniques. Springer-Verlag, 15–35

17. Rios-Bolıvar M., Zinober A. S. I. (1994) Sliding mode control for uncertainlinearizable nonlinear systems: A backstepping approach. In: Proceedings ofthe IEEE Workshop on Robust Control via Variable Structure and LyapunovTechniques, Benevento, Italy

18. Scarratt J. C., Zinober A. S. I., Mills R. E., Rios-Bolıvar M., Ferrara A., Gi-acomini L. (2000) Dynamical Adaptive First and Second-Order Sliding Back-stepping Control of Nonlinear Nontriangular Uncertain Systems. J. Dyn. Sys,Meas. and Cont. 122, 746–752

19. Yao B., Tomizuka M. (1997) Adaptive robust control of SISO nonlinear systemsin a semi-strict feedback form. Automatica. 33, 893–900

20. Yao B., Tomizuka M. (1994) Smooth adaptive sliding mode control of robotmanipulators with guaranteed transient performance. ASME J. Dyn. Syst. ManCybernetics. SMC-8, 101–109


Sliding Mode Compensation, Estimation andOptimization Methods in Automotive Control

Ibrahim Haskara1, Cem Hatipoglu2, and Umit Ozguner3

1 Visteon Corporation,Advanced Energy Transformation Sytems,36800 Plymouth Road, Livonia, MI 48150

2 Honeywell International, Bendix Commercial Vehicle Systems,901 Cleveland Street, Elyria, OH 44035

3 The Ohio State University, Department of Electrical Engineering2015 Neil Avenue, 205 Dreese Laboratory, Columbus OH 43210

Abstract. This chapter provides a broad overview of a number of recent auto-motive applications in a tutorial fashion where several analytical design tools ofthe sliding mode control theory were primarily used. The design methods are firstdiscussed from a theoretical point of view in three main categories: online func-tional optimization, disturbance/state estimation and friction compensation. Thefirst automotive control example reported in this chapter is a traction control de-sign which comprises the presented optimization and estimation methods as wellas several singular perturbation arguments. A position tracking control problem ofa throttle system which has inherent coulomb friction and stiff position feedback isthen discussed. A previous sliding mode position tracking control of a pneumaticthrottle actuator for an internal combustion engine is also summarized.

1 Introduction

Generally speaking, automotive control problems are highly nonlinear andsubject to high amount of disturbances and uncertainties. In most cases, thesystem to be controlled may operate at diverse operating regimes and includesignificant nonlinear couplings which make the abundant tools of the linearcontrol system literature not well-suited for a wide range robust operation.On the other hand, sliding mode control theory ([21], [23]) has been investi-gated in detail over the last there decades and it currently offers numeroussystematic design methods applicable to industrial control problems. The useof sliding mode control ideas in automotive control applications has also beenreported (see, for instance, [2], [3], [6], [11], [17], [24], [26]).

This chapter presents a couple recent automotive applications which blendseveral sliding mode control design methods. The theory behind the optimiza-tion, estimation and friction compensation tools used in these applicationsis first discussed in Section 2. The optimization method originates from theon-line unimodular functional optimization method of [23], [1]. The resultsof a comphrensive investigation of the use of the equivalent control idea forstate/disturbance estimation purposes are next summarized from [23], [4],


156 I. Haskara, C. Hatipoglu, and U. Ozguner

[7], [8]. The friction compensation method is based on a recent developmentwhere the handling of non-smooth nonlinearities operating on manifolds inthe state/control space is examined in a broader context via sliding motions[13], [14]. The optimization and estimation methods are then used in Sec-tion 3 on a traction control problem where the acceleration characteristicsof a vehicle are to be optimized in an engine control framework via dynamicspark advance. The friction compensation method is exemplified on a positiontracking control problem of a throttle system with inherent coulomb frictionand stiff position feedback in Section 4. Finally, Section 5 summarizes a pre-vious sliding mode position tracking control design for a pneumatic throttleactuator of an internal combustion engine.

2 Sliding Mode Design Methods

2.1 On-line Functional Optimization

To a certain extent, several automotive control objectives can be formulatedas an optimization problem. For example, ABS/traction control can be de-signed to robustly operate around the minimum/maximum point of the tireforce-relative slip curve, engine should deliver the desired torque with theleast possible fuel consumption, EGR input needs to be determined so as tominimize the emission formation and so on and so forth. The use of slid-ing modes for on-line optimization of an analytically unknown unimodularfunctional has been reported in [23]. The basic idea is to make the optimiza-tion variable (the signal which is desired to be optimized) follow an increas-ing/decreasing time function via sliding mode motions. The main difficultywith such a setup is that the unknown gradient term multiplies the controlat the differential equation of the optimization variable so that the systemitself possesses a variable structure behavior. This idea has been extended in[1] with the introduction of the notion of periodic switching function and ap-plied to ABS/traction control problems in [2], [11] and [24]. Next, we brieflydiscuss the basics of this optimization method.

Consider a unimodular functional y = f(x) which has a unique extremumat the point (x∗, y∗). The mathematical expression of f(x) is unknown. Fordefiniteness, the extremum is selected as the maximum which turns the op-timization objective into a maximization one. x is assumed to be the outputof an integrator which takes u as its input. The control objective is to keepx at the vicinity of the unknown optimal x∗ by modulating x by u usingthe on-line values of y. The performance output (optimization variable), y,is forced to track an increasing time function irrespective of the unknowngradient information via sliding modes. Pick any increasing function g(t) andtry to keep f(x) − g(t) at a constant by a proper u. If so, f(x) increases atthe same rate with g(t) independent of whether x < x∗ or x > x∗. To thisend, let

s = f(x) − g(t) (1)

Sliding Mode Compensation, Estimation and Optimization Methods 157

so that

s = (∂f/∂x) u− g(t) (2)

With the control law of

u = Msgn sin(2πs/α) (3)

as in [1] with α being a small positive constant, a sliding motion occurs forM |∂f/∂x| > |g(t)| and x is steered towards x∗ while y tracking g(t). The re-gion, defined by |∂f/∂x| < |g(t)|/M , quantifies the region in which x will beconfined with this control. The idea can be extended to more general dynam-ics by adding the derivatives of the performance variable as well as those ofg(t) to the sliding manifold expression so as to compensate the relative degreedeficit. In [12], this optimization idea has further been developed for on-lineoperating point and set point optimization purposes by ending up with atwo-time scale sliding mode optimization design. The resulting method al-lows the optimization of the closed loop operation of a system by exploitingthe extra degree of freedom in the available control authority possibly in adifferent time scale.

2.2 Disturbance Estimation and Compensation

One way of enhancing the robustness of a control system is to estimate thediscrepancies between the model used for the control derivation purposes andthe actual system by a perturbation estimator and to incorporate this infor-mation into the control law in a proper way. To this end, we next presenttwo ways of disturbance estimation. The first one is based on the equiva-lent control methodology ([23]) and it is in the continuous-time domain. Adiscrete-time sliding mode disturbance estimation method is also discussed.

First, consider a SISO nonlinear system

x = f(x) + g(x)u+ δ(t, x, u) (4)

where x ∈ R is the state, u ∈ R is the control, f(x), g(x) are smooth,known functions and δ(t, x, u) is the disturbance function which lumps allthe disturbances and the uncertainties of the system. It is assumed that

|δ(t, x, u)| ≤ ρ(t) (5)

where ρ(t) is a known bounding function. The objective is to estimate δ(t, x, u)from x.

The disturbance estimator is given by˙x = f(x) + g(x)u+ (ρ(t) + η) sgn (x− x) (6)

which basically repeats what is known about the system with an additionaldiscontinuous injection. The error dynamics follow from the subtraction ofEq. (6) from Eq. (4) as follows:

ex = δ(t, x, u) − (ρ(t) + η) sgn ex (7)


where ex = x− x. Ideally ex = 0 ∀t ≥ t0 with η > 0, x(t0) = x(t0) so that

[(ρ(t) + η) sgn ex]eq = δ(t, x, u) (8)

The operator [•]eq outputs the equivalent value of its discontinuous argumentwhich is defined as the continuous injection which would satisfy the invarianceconditions of the sliding motion (ex = 0, ex = 0) that this discontinuousinput induces. The equivalent value operator, [•]eq, can be approximatelyrealized by an high bandwidth low-pass filter according to the equivalentcontrol methodology ([23]); i.e,

τ v + v = (ρ(t) + η) sgn exv = δ(t, x, u) + O(τ, ε/τ) (9)

where |ex| ≤ ε ∀t ≥ t0 with ε being an arbitrarily small positive number.Assume that the disturbance is also differentiable; i.e,

δ(t, x, u) = ∆(t) (10)

|∆(t)| < ρ(t) (11)

where ρ(t) is a known bound. The derivative of the disturbance function canbe obtained by

˙δ(t) = K sgn (δ − δ)˙δ(t) = K sgn ([(ρ(t) + η) sgn ex]eq − δ)

where K = ρ(t) + κ, κ > 0. Therefore, in sliding mode,

[K sgn ([ (ρ(t) + η) sgn ex]eq − δ)]eq = ∆(t) (12)

The design is recursive. Equivalent control operators perform informationtransfer between two consecutive steps and the design logic can be repeatedto estimate higher order derivatives of the disturbance function as long as itis continuously differentiable to a certain order. However, the overall designrequires the implementation of the sequential equivalent value operators. Theapproximability of the equivalent control by low-pass filtering was proven in[23]. The relation between the estimation accuracy and the filter time con-stants in the implementation of the sequential equivalent value operators bylow-pass filters was examined in [8]. In that paper, an ultimate boundednessanalysis was carried out for the estimation errors and a theoretical rule ofthumb was proposed for the selection of the filter time constants.

Suppose that a baseline control law, un(t, x), has already been specifiedsuch that it would achieve the control objective for the nominal system. Thecontrol can then be complemented with the estimated disturbance as follows:

u(t, x) = un(t, x) − g−1(t, x)v (13)


where v is obtained from Eq. (9). With this new control, the closed loop dy-namics are only affected by the residual estimation error which is naturallyeasier to deal with a less conservative control action. Generally, if the distur-bance is matched with respect to the control, a direct cancellation term canbe added to the nominal control law to preserve the robustness whereas formismatched disturbances the nominal control needs to be devised so as toallow freedom for the use of the disturbance estimates. A recent study wherethese ideas were elaborated in detail can be found in [10].

Most of the today’s control algorithms are implemented in discrete-time.However, the discrete-time implementation of a continuous sliding mode con-trol law may cause the well-known chattering problem if no chattering re-duction method is employed. As an alternative estimator design where thesampling issues are taken into account at the first place, the continuous-timesystem of Eq. (14)

x = u(t) + δ(t) (14)

is first discretized so as to obtain

xk+1 = xk + Tuk + T δk (15)

where xk = x(kT ), uk = u(t) for kT ≤ t < (k + 1)T , T is the sampling time,

δk =1T

∫ (k+1)T

kT

δ(t)dt (16)

and it is assumed that the control is applied through a zero-order-hold. Notethat, δk cannot be computed unless the future values of the external distur-bance function δ(t) are known. However, if δ(t) is smooth δk can be predictedby δk−1 which can be computed from

δk =1T

[xk − xk−1] − uk−1 (17)

with an O(T ) accuracy according to

δk − δk−1 =1T

∫ (k+1)T

kT

δ(t)dt− 1T

∫ kT

(k−1)T

δ(t)dt

|δk − δk−1| = 2Tδmax = O(T ) (18)

where |dδ/dt| < δmax.Suppose that the control objective is to regulate x to zero. Incorporating

the estimated disturbance to the control law

uk = − 1Txk − δk−1 (19)

one gets

xk+1 = T [δk − δk−1] = O(T 2) (20)


so that at each sampling instant x is actually forced to an O(T 2) vicinity ofzero. This is an increase in performance compared to the direct discretizationof a discontinuous sliding mode control law which would achieve only an O(T )accuracy. A detailed study of this idea can be found in [18], [19] and [20]where it has also been shown that this control leads to an O(T 2) accuracyin sliding motion also during the inter-sampling behavior for sampled-datasystems with control being applied through a zero-order hold.

The basic idea of the discrete-time estimation method presented so far isto calculate the one step previous value of the disturbance from the availablequantities and to use it as an estimate for its current value. This results inan O(T ) estimation accuracy provided that the first order derivative of thedisturbance is bounded. If the external disturbance is continuously differen-tiable to a certain order, using more previous values of the disturbance couldactually lead to a better estimation accuracy. This idea has been studied in[5] leading to a controller structure where the ideal discrete-time equivalentcontrol law of [22] were complemented with a discrete-time filtering action toincrease the robustness of the system as well as the accuracy of the slidingmotion at the sampling instances. However, note that the intersampling timebehavior will still be O(T 2) if the control is to be implemented through a zero-order-hold as explained in [19]. Therefore, the new estimator also needs to becomplemented with an higher-order-hold mechanism in the control channelto reduce the deviations of x from zero between the two sampling instancesas well.

2.3 State Observation

In many industrial applications, the on-line estimation of several signals by anobserver rather than using a sensor may lead to more sophisticated and costeffective control systems. There are several state observer design methodsreported in the sliding mode control literature. The discussion of all thesemethods are beyond the scope of this study. Instead, in this chapter, we arespecifically interested in a sliding mode observer design method which usesthe equivalent control idea as in the continuous time disturbance estimationmethod of Section 2.2. Next this method is presented similar to its originalwhich was reported in [23]:

Consider a linear system

x = Ax+Buy = Cx

(21)

where x ∈ Rn, u ∈ Rp, y ∈ Rm, the pair (A,C) observable, C has full rank.This system can be transformed into

y = A11y +A12x1 +B1ux1 = A21y +A22x1 +B2u

(22)


where A11, A12, A21, A22, B1 B2 are constant matrices of appropriate di-mensions. The observer equation for the first part of Eq. (22) is selected asfollows:

˙y = A11y +B1u+ L1sgn(y − y) (23)

Error dynamics are given by

ey = A11ey +A12x1 − L1 sgn ey (24)

where ey = y − y. A sliding motion occurs on ey = 0 in finite time with asuitable L1 and in sliding mode

[L1 sgn ey]eq = A12x1 (25)

Therefore, an information on x1 is indirectly available through a low-passfilter.

At the second step, consider

x1 = A22x1 +A21y +B2u

y1 = L−11 A12x1

This reduced order system can also be transformed into

y1 = A31y1 +A32x2 +A33y +B3u

x2 = A41y1 +A42x2 +A43y +B4u

Let the second observer equation be

˙y1 = A31y1 +A33y +B3u+ L2 sgn(y1 − y1) (26)

Replacing y1 with its equivalent in Eq. (25) for implementation, ideally asliding motion can be guaranteed on y1 − y1 = 0 in finite time as well. Thisdesign routine can be repeated so as to result in a full finite time convergingobserver. The details and the formulatization of this method can be found in[4], [7], [8] among others. As in the disturbance estimation design, the effectsof the repeated use of low-pass filtering on the overall estimation accuracywere quantified in [8] in terms of a single variable which parameterizes allthe filter time constants. In [4], [7], a discrete-time equivalent control basedsliding mode observer design method was also proposed. Next, this methodis summarized.

Consider a discrete-time linear system

xk+1 = Φxk + Γuk

yk = Cxk(27)

where x ∈ Rn, u ∈ Rp, y ∈ Rm, the pair (Φ,C) observable and C has fullrank.


The discrete time equivalent control definition of [22] is used for a dualdiscrete-time design. Transform the original system of Eq. (27) into

yk+1 = Φ11yk + Φ12x1,k + Γ1uk

x1,k+1 = Φ21yk + Φ22x1,k + Γ2uk(28)

and let the corresponding discrete time sliding mode observer be

yk+1 = Φ11yk + Φ12x1,k + Γ1uk − vk

x1,k+1 = Φ21yk + Φ22x1,k + Γ2uk + Lvk(29)

Error dynamics are as follows:

ey,k+1 = Φ11ey,k + Φ12ex1,k + vk

ex1,k+1 = Φ21ey,k + Φ22ex1,k − Lvk(30)

where ey,k = yk − yk and ex1,k = x1,k − x1,k. The equivalent value of vk canbe calculated by solving ey,k+1 = 0 for vk ([22]) as follows:

vk,eq = −Φ11ey,k − Φ12ex1,k (31)

A sliding motion occurs on ey = 0 in finite step if vk = vk,eq. In sliding mode

ex1,k+1 = (Φ22 + LΦ12)ex1,k (32)

To implement the observer, we define an auxiliary system

zk+1 = (Φ22 + LΦ12)zk + (Φ21 + LΦ11)ey,k−1 − Ley,k (33)

and replace vk by

vk,eq = −(Φ11 − Φ12L)ey,k − Φ12(Φ21 + LΦ11)ey,k−1

−Φ12(Φ22 + LΦ12)zk(34)

By placing the eigenvalues of (Φ22 + LΦ12) at the origin, vk,eq → vk,eq andey → 0 in finite step. Note that the z-dynamics replace the low-pass filteringof the continuous-time equivalent control based observer design.

2.4 Friction Compensation

This section summarizes the theory behind the friction compensation methodof [13], [14]. To this end, we first discuss the system induced manifolds conceptand the generalized stiction phenomenon.

Consider the following class of systems

x = f (x, t) + µ · sgn (s (x, t)) + h (x, t)u (35)

with right hand-side discontinuities on the k surfaces,

s (x, t) =[s1 (x, t) , s2 (x, t) , · · · , sk (x, t)

]′= 0


where µ = [µ1 · · · µk] ∈ Rn×k, h(x, t) ∈ Rn×p, f(x, t) ∈ Rn and the entriesof f and h are smoothly differentiable functions (in Cn), u ∈ Rp is the controlinput and s : Rn → Rk. The signum operator is defined to operate on everyentry of its argument.

The system as given in Eq. (35) appears in the form of an nth-ordersystem with an input that has k + p components k of which have alreadybeen specified in the form of sliding mode control with µ and σ = x ∈Rn : s(x, t) = 0 being the gain and the manifold, respectively. Note that,this is not exactly the case as these manifolds have not been designed and theassociated gains have not been selected by the designer. Instead, they havebeen induced by the system itself. However, this sort of analogy allows us toanalyze the system using the mathematical tools of sliding mode theory.

Consider one of the candidate stiction manifolds, namely sj for j =1, · · · , k. Under the assumption of the existence of sliding mode, i.e. when

sj · sj < 0; j = 1, · · · , k (36)

the system starts to slide on the manifold described by,

σj = [x1 · · · xn−1 xn] ∈ Rn | sj(x, t) = 0 (37)

Note that the condition of Eq. (36) defines an open region Aj in the statespace. This region can be found by analyzing the derivative of sj for j =1, · · · , k

sj =d

dtsj (x1, · · · , xn−1, xn) = qj (x1, · · · , xn−1, xn) (38)

where qj ∈ R when confined to the trajectories described by Eq. (35), i.e.,when the equality in Eq. (35) is used to replace the derivatives of the statesin Eq. (38), becomes a function of the states x1, · · · , xn, the control inputu(t) and the combination of the discontinuities given on the right hand sideof the system description.

sj = qj (f1(x), · · · , fn(x), sgn(s1(x)), · · · , sgn(sk(x)),u1, · · · , up, x) + gj (x) sgn(sj(x)) (39)

where gj : Rn → R is the gain multiplying the jth discontinuous component,and qj : Rn → R. Then, (sj · sj) becomes negative if,

−gj (x, t) > |qj(·)| (40)

∀x ∈ Rn. The open region Aj is then described by,

Aj = [x1 · · · xn−1 xn] ∈ Rn | − gj (·) > |qj (·)| (41)

Recall that this analysis is prior to the controller design. When the con-dition of Eq. (40) is satisfied for some j, the system trajectories of Eq. (35)will get stuck at sj = 0 which is not a designed manifold. This phenomenon


is induced due to the inherent right hand side discontinuities existing in theoriginal system. Hence, an open stiction region can now be described in thestate space as follows,

Rs =k⋃

j=1

(σj ∩ Aj) (42)

Note that, Aj could differ from an empty set to the entire state space, butin general describes an open region which is a subset of x ∈ Rn. It is alsoaffected by the magnitude of the control input being generated.

So far, the definition of “generalized stiction” has been given. Next, asliding mode controller design approach which guarantees the avoidance ofgeneralized stiction Rs\(Rs ∩ Rc) ⊂ Rn in tracking problem for a class ofsystems where Rc is the controlled manifold.

Consider now the following class of SISO systems in their companionforms with right hand-side discontinuities on the p surfaces,

x(n) = f (x) + µ · sgn (s (x)) + h (x) v (43)v = u, y = x

where µ = [µ1 · · · µp] ∈ R1×p, h(·), f(·) : Rn → R are smoothly dif-ferentiable functions and moreover h(·) = 0 for any x = [x, x, · · · , x(n−1)]T

(controllability condition over the entire state space), u, v ∈ R, u is thecontrol input. It is allowed that there are uncertainties in f(·) and/or µ,and only some nominal values f(·) and µ are known with bounded errors∆f(·) = |f(·) − f(·)| and ∆µ = |µ− µ|. The control objective is to generatea control input u such that the output y = x tracks the reference signal xr.Define the tracking error as e = x− xr.

The controller creates multi-layer quasi-sliding manifolds so as to com-pensate for the uncertainties. To this end, let

vd =1h(·)

[x(n)

r − κ1e(n−1) − · · · − κne

]− f(·) − µ · sgnk(s(·)) + w

(44)

where w is a fictitious input, sgnk(x) = (2/π) arctan(kx) is a smooth approx-imation for the signum function with Φk(x) = sign(x) − signk(x) denotingthe approximation error.

Select σ1 = e(n−1) + c1e(n−2) + · · · + cn−2e+ cn−1e such that σ1 = 0 will

exhibit stable dynamics with negative real poles. Then pick,

w = (κ1 − c1)e(n−1) + · · · + (κn−1 − cn−1)e+ κne− β · sgnk(σ1) (45)

which will ensure (σ1 ·σ1) < 0 for all |σ1| < γ, provided that k is picked largeenough and

β > ∆f(·) + |µΦk(s(·))| + |∆µ · sgnk(s(·))| + ε (46)


Defining the sliding manifold by σ2 = v − vd and picking the correspondingcontrol input v as follows:

u = v = −α · sgn(σ2), s.t. α > (|vd| + ε) (47)

then σ2 · σ2 < 0 is ensured. In sliding mode, x → xd at the desired rate. Asmooth approximation has been used for the signum functions of the firstlayer to assure that vd is also bounded around s = 0 (which may be thecase if the signal to be tracked lies on the hyper-surface described by theinherent discontinuity or requires crossing the mentioned hyper-plane). Notethat the described control law will guarantee that the system trajectories willbe directed towards the subspace described by |σ1| ≤ γ on the n-dimensionalstate space. The magnitude of γ can be manipulated by the designer, butcannot be explicitly made zero.

3 Sliding Mode Traction Control

The acceleration characteristics of a vehicle can be improved in an enginecontrol setup where the dynamic spark advance is used to dynamically mod-ulate the engine torque [6]. The primary reason for wheel spin due to suddenchanges in the engine air input is closely related to the tire force charac-teristics of the wheel. The tire force/relative slip curve has ideally only oneextrema for each of the acceleration and the braking regions. A sufficientlylarge throttle input might cause the relative slip to move into the positivefeedback region where the tire force is decreasing with increasing slip. Sincethe availability of a relative slip measurement and an accurate analytic ex-pression of the tire force/relative slip curve are quite unrealistic in the currentsetup the sliding mode optimization method of [1] were utilized in [11], [24]to robustly operate around the peak driving tire force without any a prioriinformation on the tire force/relative slip curve.

In this section, the previous results on this topic are summarized from[11], [24]. The first control logic was to devise an optimal law for the sparkangle input in the form of an engine torque multiplier so as to keep therelative slip around its optimal which would produce the maximum tractionforce. Then, the same idea is used to optimize the performance of a baselinedynamic output feedback spark advance controller (DOFSAC) ([6]) with noadditional sensor inputs.

3.1 The Model

The plant model includes a static engine torque map, a first order trans-mission model, a nonlinear longitudinal tire force model and the vehicle isconsidered as a point mass with an aerodynamic drag force. The intake man-ifold dynamics are neglected for simplicity and no driver model is utilized.


The model equations can be written as follows:

V = a1V2 + b1Fd(t, σ)

w = a2w + a3Ψ + b2Fd(t, σ)Ψ = a4n+ a5w

n = a6n+ a7Ψ + b3Te(n,Θ)u (48)

where V is the longitudinal speed, w is the wheel speed, Ψ is a transmissionvariable, n is the engine shaft speed, σ = w/V −1 is the slip, a1 = −(Aρ/Jv),b1 = (re/Jv), a2 = −(Bw/Jw), a3 = (KTKg/Jw), b2 = −(re/Jw), a4 = 1,a5 = −Kg, a6 = −(Be/Je), a7 = −(KT /Je), b3 = (1/Je) with relevantphysical parameters and the effect of spark retard is modeled as a variableengine torque multiplier denoted by u. The value of u are then translatedinto spark timing information using an approximate static map.

The engine-wheel coupling through transmission results in a two-timescale behavior. For controller design purposes, this characteristic was utilizedto further reduce the order of the actual model based on the singular pertur-bation theory [15]. The slow system dynamics are summarized as follows:

Vs = a1V2s + b1Fd(t, σs)

ws = a2ws + b2Fd(t, σs) + a3Te(ws, Θ)us (49)

where Vs, ws, σs and us are the slow components of the variables V , w, σ andu, respectively, a2 = −(Bw + BeK

2g )/(Jw + JeK

2g ), b2 = −re/(Jw + JeK

2g ),

a3 = Kg/(Jw + JeK2g ).

3.2 Sliding Mode Dynamic Spark Advance Controller

The engine RPM and the throttle input are assumed to be available measure-ments whereas an analytical expression for the tire force/relative slip curve aswell as the optimal slip are unknown. The original control problem of robustoperation around the optimal slip is formulated as an optimization problemof an analytically unknown criterion using the optimization method summa-rized in Section 2.1. The control design is carried out on the slow system andthe tire force is obtained by the estimator of Section 2.2.

The sliding surface is selected as follows: s = e +∫ t

0Λ(e(τ))dτ where

e = Fd −F rd (t), F r

d (t) is a user-specified explicit time function and Λ(e) is tobe chosen. If s can be kept constant, the constrained motion satisfies

de

dt+ Λ(e) = 0 (50)

and the tire force behaves as desired with proper selections of F rd (t) and Λ(e).

The error variable is governed by

de

dt=

1V

∂Fd

∂σ[A(w, V, Fd) +B(w,Θ)u] +

∂Fd

∂t− dF r

d

dt(51)


where

A(w, V, Fd) = a2w + b2Fd − a1wV − b1Fdw

V(52)

B(w,Θ) = a3Te(w,Θ) (53)

Let A(•) = A + ∆A, 0 < Bmin < B(•) < Bmax where A represents thenominal part of A whereas the unknown term ∆A is bounded according to|∆A| ≤ δA with δA known and B =

√BminBmax for which β−1 ≤ (B/B) ≤ β

where β = (Bmax/Bmin)1/2.The control law is selected as u = −B−1[A + γ Φ(s)] where γ = βδA +

(β − 1)|A|∞ +M with an M > 0 and Φ(s) = sgn sin(2πs/α) is the periodicswitching function [1]. This selection guarantees that s is kept at kα for somek which depends on the system and the initial conditions, if the followingsliding mode existence condition is satisfied:

1V

∣∣∣∣∂Fd

∂σ

∣∣∣∣Mβ−1 >

∣∣∣∣∂Fd

∂t− dF r

d

dt+ Λ(e)

∣∣∣∣ (54)

If F rd is chosen as a constant, Λ(e) = λe with a λ > 0 and also assume

that explicit time dependence of the tire force is negligible, the sliding modeexistence condition becomes

1V

∣∣∣∣∂Fd

∂σ

∣∣∣∣Mβ−1 > λ |e| (55)

In sliding mode, if F rd can be reached the tire force converges exponentially

towards it with a rate dependent upon λ . On the other hand, if F rd > Fd,max

the tire force behaves as before until it enters the region where the gradientis too small such that the sliding mode existence condition of Eq. (55) canno longer be guaranteed. After that, the system becomes uncontrollable andthe tire force behaves arbitrarily. However, the controller creates a region ofattraction around the maximum point whose width can be controlled by M .Consider the region |∂Fd/∂σ| ≤ ∆. For any controller parameter M > 0,there exists a ∆ given by M > Mδ = V λ|Fd − F r

d |max/∆ such that the tireforce is guaranteed to be kept in this region.

3.3 Optimal Sliding Mode DOFSAC

The original DOFSAC ([6]) controls the engine torque output via dynamicspark advance based on filtered engine RPM measurement. Engine RPM isfiltered with a band pass filter for practical differentiation purposes and thenit is compared to a constant threshold value. If the filter output is greaterthan the threshold, the spark timing is retarded proportional to the error.This decouples the high energy terms of the engine from the wheel so thatthe likelihood of wheel spin reduces. However, the threshold value needs to beselected and originally it is tuned in advance for different conditions throughsimulation studies and experimental tests.


Vehicle

SLIDING MODE

OPTIMIZER

(s)DifferentiatorK1- u

Engine Torque

Multiplier

Throttle Angle

Engine RPM

Set Point

0.5

0.5

Σ+

−

Fig. 1. Optimized DOFSAC (from [11], [24])

This section summarizes an optimal sliding mode DOFSAC design wherebasically an additional loop is devised for the threshold so as to maximize thetire force (Fig. 1). To this end, DOFSAC is first modeled by u = 1−K(n−ρ) =ρ−Kn where ρ denotes the threshold and K is selected such that the errorK(n− ρ) is properly mapped to a value in the admissible control domain toproduce a control within its limits. Using the singular perturbation theory,the order of the complete system with the control in the loop can also bereduced as in Section 3.2 as follows:

Vs = a1V2s + b1Fd(t, σs)

ws = a2ws + b2Fd(t, σs) + a3Te(ws, Θ)ρs (56)

with state dependent a2, b2 and a3. Repeating the design of Section 3.2, theset point is obtained from ρ = −B−1[A + γ Φ(s)] where s = e +

∫ t

0Λ(e)dτ ,

Λ(e) is to be chosen and γ = βδA + (β − 1)|A|∞ + M . With a sufficientlylarge M , this set point selection forms a positively invariant region aroundthe optimal slip defined by |∂Fd/∂σ| ≤ ∆. In sliding mode, e + Λ(e) = 0and the tire force converges to this region as desired with proper F r

d andΛ(e). The size of this region can also be controlled by M . Once ρ has beendetermined ρ can be computed using ρ = K−1(ρ− 1). Further details of thepresented traction control designs as well as the simulation results can befound in [11].


4 Friction Compensation for the Position Control of aThrottle System

The system involves a plant which is driven by an actuator with faster dy-namics. The plant has inherent coulomb-viscous friction and stiff positionfeedback which are the two sources of stiction in the state space.

+ +1 1 11f ( )

xx.

z z+

2

h( x , z )

u ω ωω

Coulomb Friction

Stiff Position Feedback

Plant

Actuator..

s s

f (h(.), )3 ω

s

f ( x )

Fig. 2. The system to be controlled (from [14])

4.1 The Plant Model

Consider the plant depicted in Fig. 2. The state space representation is givenby

x = (1/Kg)f1(ω) (57)ω = (1/J) (h(x, z) − (C/Kg)ω − (1/Kg)f3(h(x, z), ω))z = (1/L) (−Rz −Ktω + u)

where

f1(ω) = deadzone(ω,±δ, 1) (58)

h(x, z) = Ktz − (1/Kg) (f2(x) +K (180x/π − θo)) (59)

f2(x)! = γsat ((180x/π − θo)α) (60)

f3(h(.), ω) = βsat (f1(ω)/δ) + βsat(Kgh(.)f4(f1(ω))) (61)

f4(f1(ω)) = 1− (relaywdzn(f1(ω),±δ, 1))2 (62)

where x is the position, ω is the angular velocity, and z is the auxiliary statevariable that describes the dynamics of the first order actuator. Due to thephysical limits of the system the position variable x(t) should be between7 and 85. The referred nonlinearities, the desired tracking signals and theparametric values can be found in [14].


4.2 An Approximate Model for Control Design

Although friction has been modeled in details so as to include the Stribeckeffects as well as stick-slip behavior, it is concluded that a simpler modelsuffices to describe the motion of the system with good precision while easingthe controller design phase. Consider,

x = a12ω (63)ω = a21(x− xo) + a22ω − κsgn (x− xo) − µsgn(ω) + a23z

z = a32ω + a33z + bu

where a12 = (1/Kg), a21 = −(K/KgJ), a22 = −(C/KgJ), a23 = (Kt/J),a32 = −(Kt/L), a33 = (R/L), b = (1/L), xo = (πθo/180), κ = (γ/KgJ)and µ = (β/KgJ). The unforced system (when u = 0) converges the stableequilibrium point given by (x, ω, z)eq = (x∗o, 0, 0) where x∗o = x ∈ R : |x−xo| ≤ ζ in the sliding mode sense starting from any initial conditions due tothe existence of the discontinuous terms on the right hand side of the statespace representation in Eq. (63). Based on the numerical data, it has beenobserved that the coupling on the third equation is weak so that the z termin the second equation can be replaced with z = −(a32/a33)ω according tothe singular perturbation theory.

4.3 Controller Design

Let ex = x − xr where xr is the reference to be followed. From (63), oneobtains,

x = a12 (a21(x− xo) + (a22/a12)x+ · · · (64)· · · −κsgn(x− xo) − µsgn (x/a12) + a23z)

Following the design method of Section 2.4, the control law is governedby

zfl = zfl + (w/a12a23) (65)

zfl =1

a12a23[xr + ξ1xr + ξ2xr + a12a21xo+ (66)

· · · − (a12a21 + ξ2)x− (a22 + ξ1)x+· · · + a12κsgn(x− xo) + a12µsgn(x/a12)]

sw = ex + C1exw = (ξ1 − C1)ex + ξ2ex − Msgnk(sw)

sz = z − zfl

u = −Msgn(sz)

(67)


where C1 > 0, M > |Φ(·)| with Φ(·) being the lumped uncertainty origi-nating from the bounded uncertainties in the plant parameters, M is suffi-ciently large positive number so as to induce a sliding motion on sz = 0. Thespeed information required for the control implementation is obtained by anequivalent control based observer whose design idea has been presented inSection 2.3. The details of the overall control design summarized above andthe simulation results can be found in [14].

5 Position Control of a Throttle System

This section presents a previous throttle angle position controller developedat the Ohio State University as a part of an Intelligent Vehicles and HighwaySystems study. The existing pneumatic throttle actuator of a 1992 HondaAccord station-wagon was controlled for vehicle speed control purposes.

5.1 Throttle Actuator Model

AtmosphericPressure

Vent

Safety

Vacuum

AB

C

ThrottleCable

Air Cylinder

Solenoids

Intake ManifoldPressure

AcceleratorPedalInterface

ThrottlePlate

Torsional Spring

TorsionalSpring

AcceleratorPedal

LeverArm

Throttle Actuator

Fig. 3. The pneumatic throttle system (from [16]

The system includes the throttle actuator, throttle cable, and the throt-tle plate connection (Fig. 3). The throttle actuator is a pneumatic cylinderwhich creates a force proportional to the ratio of the cylinder’s internal airpressure to the external (atmospheric) air pressure. The internal air pressureis controlled using two valves which allow either the engine’s intake mani-fold pressure or the atmospheric pressure be applied to the input of the aircylinder. One cable connects the pneumatic cylinder to the accelerator pedalwhile a second cable connects the accelerator pedal to the throttle plate. Theactuator’s internal pressure is controlled using three solenoid actuated valveswhich control the air flow in and out of the pneumatic cylinder. The throttleangle is controlled by opening or closing the vent and vacuum valves until theinternal air pressure that is needed to move the throttle angle to the desiredposition is achieved.


For control law derivation purposes, two separate second order linear mod-els were experimentally determined, one of which is valid when the vent valveis open and the other when in full vacuum mode. If both the vent and vacuumvalves are closed the system is assumed to behave according to the unforcedvacuum model.

5.2 Sliding Mode Control of the Throttle Angle

Due to the nature of the control input to the throttle actuator system beingeither full vacuum or full vent, a sliding mode design was adopted. The slidingsurface was defined as s = e+ ke with e = θ− θdes, θ is the throttle angle indegrees and θdes was the desired throttle angle. u takes two values, +1 and−1, which represent full vacuum or full vent, respectively, depending on s.

The region in which the sliding mode existence condition can be guaran-teed to hold with the available control authority was determined by consid-ering the worst-case scenarios. This region is clearly affected by the value ofk. For implementation, k was selected to have a reasonable decay in slidingmode by giving up the global sliding mode existence although it was possibleto select a k which would provide the control objective globally. In order toimplement the control, θ, which was not directly available, was estimatedby a linear Kalman filter. Since there were two possible linear systems, thefilter parameters were determined individually and switched according to theinput. The further details of the design as well as the experimental resultswere reported in [16].

6 Concluding Remarks

The usage of sliding mode estimation, optimization and compensation meth-ods in automotive control problems have been demonstrated on there differ-ent examples: a traction control design for anti-spin acceleration, a trackingcontrol design for a throttle system subject to stiction nonlinearities and aposition tracking control design for a pneumatic throttle system of an internalcombustion engine. The theoretical background and the relevant literature onthe sliding mode design methods used have also been reported for an easyreference.

References

1. Drakunov, S.V., Ozguner, U. (1992) Optimization of nonlinear system outputvia sliding mode approach. Proceedings of the IEEE International Workshopon Variable Structure and Lyapunov Control of Uncertain Dynamical Systems,Sheffield, UK, 61–62

2. Drakunov, S. V.,Ozguner, U., Dix, P. and Ashrafi, B. (1995) ABS control us-ing optimum search via sliding modes. IEEE Transactions on Control SystemsTechnology, 3, 79–85


3. Drakunov, S. V., Hanchin, D., Su, W-C., and Ozguner, U. (1997). Nonlinearcontrol of a rodless pneumatic servoactuator or sliding modes versus coulombfriction. Automatica 33, 7, 1401–1406

4. Haskara, I., Ozguner, U., and Utkin,V. I. (1996) On variable structure observers.Proceedings of the IEEE International Workshop on Variable Structure Systems,Tokyo, Japan, 193–198

5. Haskara, I., Ozguner, U., and Utkin,V. I. (1997) Variable structure control foruncertain sampled data systems. Proceedings of the 36th Conference on Decisionand Control, San Diego, CA, 3226–3231

6. Haskara, I., Ozguner, U., and Winkelman, J. (1998) Dynamic spark advancecontrol. IFAC Workshop-Advances in Automotive Control Preprints, 249–254

7. Haskara, I., Ozguner, U., and Utkin,V. I. (1998) On sliding mode observers viaequivalent control approach. International Journal of Control, 71, 6, 1051–1067

8. Haskara, I., Ozguner, U. (1999) Equivalent value filters in disturbance estima-tion and state observation. Variable structure systems, sliding mode and non-linear control. K.D. Young and U. Ozguner eds., Lecture Notes in Control andInformation Science, 247, 167–179, Springer Verlag

9. Haskara, I. (1999) Sliding mode estimation and optimization methods in non-linear control problems. Ph.D. Thesis, The Ohio State University, Columbus,OH

10. Haskara, I., Ozguner, U. (1999) An estimation based robust tracking controllerdesign for uncertain nonlinear systems in strict feedback form. Proceedings ofthe 38th Conference on Decision and Control, Phoenix, AZ

11. Haskara, I., Ozguner, U., and Winkelman, J. (2000) Wheel slip control forantispin acceleration via dynamic spark advance, IFAC Journal of Control En-gineering Practice, 8, 10, 1135–1148

12. Haskara, I., Ozguner, U., and Winkelman, J. (2000) Extremum control foroptimal operating point determination and set point optimization via slidingmodes. Transactions of the ASME, Journal of Dynamic Systems, Measurement,and Control, 122, 4, 719–724

13. Hatipoglu, C. (1998) Variable structure control of continuous time systemsinvolving non-smooth nonlinearities. Ph.D. Dissertation, The Ohio State Uni-versity, Columbus, OH

14. Hatipoglu, C., Ozguner, U. (1999) Handling stiction with variable structurecontrol. Variable structure systems, sliding mode and nonlinear control. K.D.Young and U. Ozguner eds., Lecture Notes in Control and Information Science,247, 143–166, Springer Verlag

15. Kokotovic, P. V., Khalil, H. K., and O’Reilly, J. (1986) Singular PerturbationMethods in Control: Analysis and Design. Academic Press, London.

16. Sommerville, M., Hatipoglu, C., and Ozguner, U. (1996) On the variable struc-ture control of a throttle actuator for speed control applications. Proceedings ofthe IEEE International Workshop on Variable Structure Systems, Tokyo, Japan,187–192

17. Sommerville, M., Hatipoglu, C., and Ozguner, U. (1998) Switching control ofa pneumatic throttle actuator. IEEE Control Systems Magazine, 81–87

18. Su, W-C., Drakunov, S. V., and Ozguner, U. (1993) Sliding mode control indiscrete time linear systems. Preprints of IFAC 12th World Congress, Sydney,Australia.

19. Su, W-C. (1995) Implementation of variable structure control for sampled-datasystems. Ph.D. Dissertation, The Ohio State University, Columbus, OH


20. Su, W-C., Drakunov, S. V., and Ozguner, U. (1996) Implementation of variablestructure control for sampled-data systems. Robust Control via Variable Struc-ture and Lyapunov Techniques, F. Garofalo and L. Glielmo eds., Lecture Notesin Control and Information Sciences Series, 217, 87–106, Springer-Verlag

21. Utkin, V. I. (1977) Variable structure systems with sliding modes. IEEE Trans-actions on Automatic Control, 22, 2, 212–222

22. Utkin, V., Drakunov, S. (1989) On discrete-time sliding mode control. Proceed-ings of IFAC Symposium on Nonlinear Control Systems (NOLCOS), 484–489

23. Utkin, V. I. (1992) Sliding Modes in Control and Optimization. Springer-Verlag24. Winkelman, J., Haskara, I., and Ozguner, U. (1999) Tuning for dynamic sparkadvance control. Proceedings of American Control Conference, 163–164, SanDiego, CA

25. Young, K. D., Ozguner, U. (1993) Frequency shaping compensator design forsliding mode. International Journal of Control, 57, 5, 1005–1019

26. Young, K. D., Ozguner, U.(1997) Sliding mode design for robust linear optimalcontrol. Automatica, 33, 7, 1313–1323

27. Young, K. D., Utkin, V.I., and Ozguner, U. (1999) A control engineer’s guideto sliding mode control,” IEEE Transactions on Control Systems Technology, 7,3, 328–342

On Quasi-optimal Variable Structure ControlApproaches

Jian-Xin Xu and Jin Zhang

E.C.E. Dept., National University of Singapore, Singapore 117576

Abstract. In this paper, variable structure control (VSC) approaches are inte-grated with nonlinear optimal control approaches. VSC design consists of twophases: sliding mode design and switching control design. For the first phase, wepropose a nonlinear sliding mode design methodology incorporating optimality fora class of nonlinear MIMO systems. Two issues are addressed in detail: 1) how toconstruct a sliding mode for cascaded nonlinear dynamics where the linear-typesliding mode design is not applicable; 2) how to achieve a nonlinear sliding modewith optimality. As for the second phase, a kind of nonlinear suboptimal controlaccording to the system nominal part is integrated with VSC mechanism with thepredesigned nonlinear sliding mode. By integration, we achieve a quasi-optimal con-troller in which the suboptimal control part and VSC part are made to functionin a complementary manner. In particular when the system nominal part is pre-dominant, the nonlinear optimal control part will govern the system response aswell as drive the system to approach the equilibrium in an optimal fashion. Onthe contrary, when the system perturbation becomes the main factor, the VSC willtake over the control task to warrant the desired control precision by its robustnessproperty.

1 Introduction

Variable structure control (VSC) has been widely recognized as a powerfulcontrol strategy for its ability of making a control system very robust. Nu-merous theoretical studies as well as application research have been reported[1–7].

A typical VSC design consists of two phases, the first phase is the slidingmode design. In [2] a linear-type optimal sliding mode σ is designed for thelinear cascaded systems

x1 = A11(t)x1 +A12(t)x2

x2 = A21(t)x1 +A22(t)x2 +B(t)(u + d)

where d is the matched system uncertainty. Note that the selected slidingmode must stabilize the sliding manifold described by

σ = Cx1 + x2 = 0x1 = A11(t)x1 +A12(t)x2,

which leads to well known pole-placement problem for the x1-subdynamics

x1 = (A11 −A12C)x1.


A LQR approach was suggested in [2] to conduct a systematic design for theMIMO x1-subdynamics and meanwhile achieve the optimality for the slidingmanifold.

A limitation of the linear-type sliding mode is that they can hardly han-dle nonlinear x1-subdynamics. Pole-placement or LQR approaches, thoughare very effective in linear systems, may not be valid in the nonlinear world.Linear approximation of the nonlinear x1-subdynamics is only valid locallyaround the equilibrium where it is linearized. Consider a nonlinear x1 - sub-dynamics still with the linear sliding mode

σ = Cx1 + x2 = 0x1 = f(x1,x2, t).

In general there may not exist such a matrix C that can stabilize the non-linear sliding manifold x1 = f(x1,−Cx1, t). We have to take the systemnonlinearites f into account in the sliding mode design, and in the sequelcome up with a nonlinear sliding mode. Analogous to [2], another importantissue in nonlinear sliding mode design is, whether optimality can be pursued.

In this paper, we propose a systematic sliding mode design methodologyfor a class of nonlinear systems, which is able to take into account the non-linearities in x1-subdynamics and incorporate optimality. Optimal controlnot only provides a systematic design approach for MIMO systems but alsoprovides extra degrees of freedom to specify the system performance withselected cost functions. In this work we consider three cases. A nonlinearoptimal sliding mode can be constructed if the HJB (Hamilton-Jacobian-Bellman) equation is solvable. In particular if the x1-subdynamics is non-linear only in inputs (x2), a nonlinear optimal sliding mode can be easilydesigned by solving an algebraic Ricatti equation. If however the HJB is notsolvable but there exists a static stabilizing control law for nonlinear x1 dy-namics, a nonlinear sliding mode with inverse optimality can be constructedaccordingly.

The second phase is the switching control design. In VSC design, the sys-tem nominal part is usually canceled out. Many real physical systems aremore or less transparent to us with only a small portion of uncertainties,and the nominal part dominates system response. When the system nominalpart is dominant, robustness is no longer the only concern of control designand other performance requirements should be taken into consideration. Aninteresting question is, while retaining the system stability of VSC, can weintroduce aother performance index, such as minimizing input energy, achiev-ing faster tracking convergence, etc.? The answer is yes because we can applyoptimal control design to the system nominal part, and apply VSC only tonorm-bounded uncertainties. Consider a special nonlinear x2 dynamics (thegeneral one will be presented in subsequent sections)

x2 = f + u + d.

176 J.-X. Xu and J. Zhang

where f is the nominal part. An optimal control uop can be designed accord-ing to f , the VSC part uvsc according to d, and both work concurrently asu = uop + uvsc. It is obvious that, in the region dominated by the systemnominal part, i.e. ‖f‖ ‖d‖, the system behavior is mainly governed byoptimal control. On the contrary, in the region where perturbation becomesdominant, i.e. ‖d‖ > ‖f‖, VSC will naturely take over the main control task.A problem encountered in uop design is the difficulty in sloving the HJBequation. To facilitate optimal design, a suboptimal nonlinear control designis considered [10]. The integration of the nonlinear optimal sliding mode, sub-optimal control and VSC results in a new control approach – quasi-optimalVSC, which offers the appreciated robustness property, optimality in con-trol performance, and systematic design for quite general classes of nonlineardynamic systems.

The organization of this chapter is as follows. In Sec. 2, the typical VSCin the presence of system uncertainties is presented. In Sec. 3, nonlinear timevarying sliding modes with optimality are constructed based on nonlinearoptimal control, the algebraic Ricatti equation and inverse optimal controlrespectively. In Sec. 4, the suboptimal controller based on CLF is illustrated.In Sec. 5, the VSC and the suboptimal control are integrated. In Sec. 6,illustrative examples are presented to demonstrate the validity of the newidea and effectiveness of the proposed approach.

Nomenclature

In this Chapter the following nomenclature will be adopted.

V := Lyapunov function;V c := control Lyapunov function (CLF);

V1(x1) := the first part of V c, a positive definate function of x1;V ∗ := optimal value functionσ := switching surface of VSC;

σ1 := the first part of σ, a function of x1 and t;Q,R := weighting matrices in the performance index of the nonlinear

optimal sliding mode;l(x1) := penalty of system states x1 of the nonlinear optimal sliding

mode;q(x) := penalty of system states x1 and x2 of the suboptimal control

part of quasi-optimal VSC;u := control input;

uvsc := VSC (switching) part of u;uc := compensation part of VSC law;

177On Quasi-Optimal Variable Structure Control Approaches

uop := suboptimal control law or the suboptimal control part of thequasi-optimal VSC law;

u∗ := optimal control law;v := virtual control input to the x1-subsystem;

v∗ := virtual optimal control input to the x1-subsystem

2 Typical VSC System Construction

Consider the following nonlinear uncertain MIMO system

x1 = f1(x1, t) +G1(x1, t)ϕ(x1,x2, t)x2 = f2(x1,x2, t)

+B2(x1,x2, t)[I +∆B2(x1,x2, t)]u + d(x1,x2, t)(1)

where x1 ∈ Rn1 and x2 ∈ Rn2 are the physically measurable state vectors.u ∈ Rn2 is the control input. ϕ ∈ Rn2 . ∂ϕ

∂x1and ∂ϕ

∂x2are bounded and

∂ϕ∂x2

= 0 in D ⊂ Rn1⋂Rn2

⋂[0,∞). f1, f2, ϕ, G1 and B2 are the known

nominal part of the system with appropriate dimensions. G1 is continuouslydifferentiable in all arguments. d and ∆B2 represent system additive andmultiplicative input uncertainties respectively. I denotes an identity matrixwith appropriate dimensions.

Assumption 1: ∀(x1,x2, t) ∈ D,

‖∆B2‖ =√λmax(∆BT∆B) ≤ εb 0 < εb < 1

‖d‖ =√d21 + d2

2 + · · ·+ d2n2≤ρm(x1,x2, t)

where ρm is a known positive upper bounding function.

Assumption 2: ∀(x1,x2, t) ∈ D, ∂ϕ∂x2

B2 is full rank.One of the most effective ways to control this kind of systems is VSC

which ensures the stability in the presence of system perturbations d and∆B2.

To construct a VSC, first choose a nonlinear switching surface

σ = σ1(x1) + σ2(x2) = σ1(x1) + x2. (2)

Then choose a positive definite function V = 12σT σ and differentiate it, we

have

V = σT σ

= σT [∂σ1

∂x1I]

[f1 +G1ϕ

f2

]+B2[(I +∆B2)u + d]

= σTB2B−12 [

∂σ1

∂x1I]

[f1 +G1ϕ

f2

]+ (I +∆B2)u + d

= αT [−uc + (I +∆B2)u + d], (3)


where

αT = σTB2 (4)

uc= −B−1

2 [∂σ1

∂x1I]

[f1 +G1ϕ

f2

].

Letting u = uc + uvsc and substituting it into (3), yields

V = αT [∆B2uc + d + (I +∆B2)uvsc]= αT (∆B2uc + d) + αT uvsc + αT∆B2uvsc

≤ ‖∆B2uc + d‖ · ‖α‖+ αT uvsc + αT∆B2uvsc. (5)

Define

ρv(x, t) = εb‖uc‖+ ρm(x, t) ≥ ‖∆B2uc + d‖and choose

uvsc = − [ρv(x, t) + δ]α(1− εb)‖α‖ ,

where δ > 0 is a constant. By substituting uvsc into (5), we have

V ≤ ρv(x, t)‖α‖ − αT [ρv(x, t) + δ]α(1− εb)‖α‖ + εb

‖αT ‖[ρv(x, t) + δ]‖α‖(1− εb)‖α‖

= ρv(x, t)‖α‖ − ρv(x, t) + δ

(1− εb)‖α‖+ εb

ρv(x, t) + δ

(1− εb)‖α‖

= −δ‖α‖.Thus the VSC law

u = uc + uvsc

= −B−12 [

∂σ1

∂x1I]

[f1 +G1ϕ

f2

]− [ρv(x, t) + δ]α

(1− εb)‖α‖ (6)

leads to

V ≤ −δ‖α‖ < 0. (7)

The negative definiteness of V implies a finite reaching time to the switchingsurface σ = 0.

When the system is in the sliding mode, the stability of the sliding man-ifold is jointly determined by σ1 and the x1-subdynamics. However, it is ingeneral a difficult task to choose σ1 which stabilizes the sliding manifold forthe x1-subdynamics given in (1). Besides, the system nominal part is simplycanceled out by uc in the above VSC design. It would be highly preferred ifthis part of the system knowledge can be better made use of in VSC design.


3 Sliding Mode Design with Optimality

In this section, we propose a systematic nonlinear sliding mode design strat-egy with optimality. The underlying idea is to convert the sliding mode designinto an optimal control design for the x1-subsystem. In this way not only canwe design a nonlinear sliding mode systematically, but also acquire the de-sired optimality.

3.1 Nonlinear Optimal Sliding Mode Design

Look at system (1), we can treat ϕ(x1,x2, t)= v as a virtual control input to

the x1-subsystem, though the actual input is x2. Now consider the followingoptimal control task

x1 = f1(x1) +G1(x1)v

infv(·)

∫ ∞

0

[l(x1) + vTRv]dt (8)

where R = RT > 0 and l(x1) > 0.The HJB equation, a standard method to solve optimal problems [9], of

this system is

V ∗x1

f1 −14V ∗

x1G1R

−1GT1 V

∗Tx1

+ l(x1) = 0 (9)

where V ∗x1

= [∂V ∗∂x1

]Tand V ∗ is commonly referred to as the value function

and can be thought of as the minimum cost to go from the current statex1(t), i.e.,

V ∗(x1(t)) = infv(·)

∫ ∞

t

[l(x1(τ)) + vT (τ)Rv(τ)]dτ.

Suppose that there exists a C1 positive semidefinite function V ∗(x1) whichsatisfies the HJB equation (9), then the optimal control law is given by

v∗(x1) = −12R−1GT

1 V∗Tx1

(x1).

Now define

σ1(x1) =12R−1GT

1 V∗Tx1

(x1)

= −v∗(x1). (10)

Theorem 1. The following nonlinear sliding mode

σ(x1,x2, t) = σ1(x1, t) + σ2(x1,x2, t) = σ1(x1) + ϕ(x1,x2, t) = 0 (11)

where σ ∈ Rn2 , is an optimal sliding mode for system (1) with σ1 defined in(10).


Proof. Note that the x1-subdynamics is as follows

x1 = f1(x1) +G1(x1)ϕ(x1,x2, t).

When the system is in the sliding mode defined in (11), we have

ϕ(x1,x2, t) = −σ1(x1) = −12R−1GT

1 V∗Tx1

(x1)

= v∗(x1)

This shows that the nonlinear sliding mode (11) does lead to a nonlinearoptimal control law for the optimal control task (8). As a result, the opti-mality of the nonlinear sliding mode ensures the asymptotic stability of thenonlinear sliding manifold

x1 = f1(x1) +G1(x1)ϕ(x1,x2, t)

= f1(x1)− 12G1(x1)R−1GT

1 V∗Tx1

(x1).

Remark 1. In calculating the nonlinear sliding mode we fully use the knowl-edge of the system nonlinearities. It is not necessary to solve (11) to get theexplicit expression for x2. Thus ϕ can be non-affine in x2.

3.2 Nonlinear Optimal Sliding Mode Design with InputNonlinearity

In the above optimal sliding mode design, in order to solve the optimal controltask (8) we need to find the solution of the HJB partial differential equationwhich may not be feasible for many nonlinear systems. However, if f1 andG1 in (1) are linear or linearizable, then the x1-subdynamics in (1) can berewritten as

x1 = f1 +G1ϕ(x1,x2, t) = A11x1 +A12ϕ(x1,x2, t), (12)

where (A11, A12) is controllable. Note that the input to the x1-subdynamicsmay have nonlinear factors. In general ϕ(x1,x2, t) can be affine or non-affine in x2, such as x2e

−x2 in scalar case. Regardless of the presence ofnonlinearities, we can construct a nonlinear optimal sliding mode for system(12) through solving the algebraic Ricatti equation analogous to the linearcases.

Consider the optimal control task below,

x1 = A11x1 +A12v

infv(·)

∫ ∞

0

[xT1 Qx1 + vTRv]dt (13)

where Q = QT > 0 and R = RT > 0. The optimal control law of this linearsystem is given by

v∗(x1) = −R−1AT12Px1,


where P is the solution of the algebraic Ricatti equation

PA11 +AT11P − PA12R

−1AT12P +Q = 0

and the optimal value function is

V ∗ = xT1 Px1.

Analogous to the sliding mode construction (11) in the previous subsection

σ = σ1(x1) + σ2(x1,x2) = Cx1 + ϕ(x1,x2, t) = 0 (14)

with

C = R−1AT12P.

This nonlinear sliding mode ϕ(x1,x2, t) = −R−1AT12P yields the optimal

control law for x1-subdynamics (12). As a consequence the resulting nonlinearsliding manifold is asymptotically stable.

Remark 2. If ϕ(x1,x2, t) is simply a linear vector x2, and R in the costfunction (13) is chosen to be a high pass filter, we can easily reach a frequencyshaped optimal sliding mode [2].

3.3 Sliding Mode Design with Inverse Optimality

If the x1-subdynamics in (1) cannot be simplified into the linear case (12) orthe HJB equation (9) is not solvable, we may look for an alternative way –constructing a nonlinear sliding mode with inverse optimality. The underlyingidea is as follows. First design a static control law with respect to a Lyapunovfunction. Then an optimal control law associated with a specific cost functioncan be constructed accordingly, which lead to a solution to a specific HJBequation. In the sequel the optimal control law can be used to construct thenonlinear optimal sliding mode.

Rewrite the optimal control task (8),

x1 = f1(x1) +G1(x1)v

infv(·)

∫ ∞

0

[l(x1) + vTRv]dt. (15)

Assume there exists a static state feedback control law

v = −R−1G1(x1)TV Tx1(x1) (16)

stabilizes the x1-subdynamics with respect to a positive definite and radiallyunbounded Lyapunov function V (x1), i.e.

V = Vx1 x1 = Vx1f1 + Vx1G1v < 0.


Then the control law

v∗ = 2v = −2R−1G1(x1)TV Tx1(x1)

is optimal with respect to the performance index (15) where

l = −4(Vx1f1 + Vx1G1v)= −4(Vx1f1 − Vx1G1R

−1GT1 V

Tx1). (17)

Note that from V , ∀ x1 = 0,

Vx1f1 + Vx1G1v < 0,

thus l(x1) is positive definite for all x = 0 and can be chosen as part of thecost function. Now let us show that v∗ and l(x1) satisfy the HJB equation(9) with the optimal value function

V ∗ = 4V . (18)

This can easily be verified below

V ∗x1

f1 −14V ∗

x1G1R

−1GT1 V

∗Tx1

+ l(x1)

= 4Vx1f1 +14· 4Vx1G1R

−1GT1 · 4V T

x1− 4(Vx1f1 − Vx1G1R

−1GT1 V

Tx1)

= 4Vx1f1 + 4Vx1G1R−1GT

1 VTx1

− 4Vx1f1 − 4Vx1G1R−1GT

1 VTx1

= 0.

Using the above inverse optimal approach we can now define the nonlinearoptimal sliding mode in a similar way as in (11)

σ(x1,x2) = σ1(x1) + σ2(x1,x2) = σ1(x1) + ϕ(x1,x2, t) = 0

with

σ1 = −2v = 2R−1G1(x1)TV Tx1(x1)

which ensures asymptotic stability and optimality of the nonlinear slidingmanifold.

Remark 3. The inverse optimal approach illustrated in this section rendersthe task of solving the HJB partial differential equation (a tougher one) intothe task of looking for a stabilizing control law (16) in conjunction with aLyapunov function V (x1) (a relatively easy one), and still provides certaindegrees of freedom in the optimal controller design, such as the selection ofthe weighting matrix R.


3.4 Robustness Analysis in the Presence of Sector Uncertainties

In sliding mode design, whether for linear or nonlinear cases, it is necessaryto know the x1-subdynamics a priori. If there exists uncertainties in the x1-subdynamics, robustness should be taken into consideration in the slidingmode design. In the linear case, the optimal sliding mode using LQR designachieves the infinity gain margin and 60 degrees phase margin. Similarly, thenonlinear optimal sliding mode, based on nonlinear optimal control design[9], achieves a sector margin (1

2 ,∞).Consider the following uncertain MIMO system

x1 = f1(x1, t) +G1(x1, t)ϕ(x2, t)x2 = f2(x1,x2, t)

+B2(x1,x2, t)[I +∆B2(x1,x2, t)]u + d(x1,x2, t),(19)

with sector-type uncertainty in ϕ(x2, t), i.e. ϕ(x2, t) = [ϕ1, ϕ2, · · · , ϕn2 ]T and

ϕi, i = 1, 2, · · · , n2 are unknown but belong to a sector (12 ,∞)

12x2 < xϕi(x, t) < ∞.

By virtue of the sector margin property of nonlinear optimal control, we canconstruct a stable sliding mode for system (19), as shown by the followingtheorem.

Theorem 2. Construct the following sliding mode

σ = σ1(x1) + σ2(x2) = σ1(x1) + x2 = 0, (20)

σ1(x1) =12R−1GT

1 V∗Tx1

(x1)

where R = diag(r1, · · · , rn2) > 0 and V ∗ is the solution of the HJB equation(9). This sliding mode results in an asymptotically stable sliding manifold

x1 = f1(x1, t) +G1(x1, t)ϕ(−σ1) (21)

where ϕ(·) is the sector-type uncertainty.

Proof. Choose a positive definite Lyapunov function

V (x1) = V ∗(x1),

differentiate it and use the HJB equation (9), we have

V = V ∗x1[f1 +G1ϕ(x2)]

= −l(x1) +14V ∗x1G1R

−1GT1 V

∗Tx1

+ V ∗x1G1ϕ(x2).

When the system is in the sliding mode σ = 0

x2 = −σ1 = v∗(x1) = −12R−1GT

1 V∗Tx1

(x1).


Note the sector-type uncertainty satisfies ϕ(−x2) = −ϕ(x2). Thus

V = −l(x1) +12V ∗

x1G1σ1 + V ∗

x1G1ϕ(−σ1)

= −l(x1)− V ∗x1G1[ϕ(σ1)− 1

2σ1]

= −l(x1)− 2σT1 R[ϕ(σ1)− 1

2σ1]

< −2σT1 R[ϕ(σ1)− 1

2σ1].

Note that R = diag(r1, r2, · · · , rn2) > 0 for x1 = 0 and ϕi(·) belongs to asector ( 1

2 ,∞). Therefore,

V < −2n1∑i=1

riσ1,i[ϕi(σ1,i)− 12σ1,i] < 0.

Remark 4. The sliding mode (20) constructed in theorem 2 can guaranteethe asymptotical stability of the x1-subdynamics, but may not be optimal.

4 Construction of Nonlinear Suboptimal ControlBased on Nominal System and CLF

It has been shown that the system nominal part is canceled out in the typicalVSC law (6) by uc. Instead of cancelation, now let us explore the possibilityof constructing an optimal controller for the system nominal part.

The nominal part of system (1) is

x = f +Bu (22)

where x = [xT1 ,x

T2 ]

T , f = [ [f1 +G1ϕ]T , fT2 ]

T , B = [0, BT2 ]

T . Consider theperformance index

infu(·)

∫ ∞

0

[q(x) + uT u]dt. (23)

Usually it is very hard or impossible to find an optimal controller associatedwith (22) and (23). Thus, we consider a suboptimal control method based onSontag’s formula [10]. The Sontag’s formula for the above problem is

uop =

−V c

x f+√

[V cx f ]2+q(x)[V c

x BBT V cTx ]

V cx BBT V cT

xBTV cT

x V cxB = 0

0 V cxB = 0

(24)

where V c is a Control Lyapunov Function for the nominal part of system (1).

Definition 1. According to [9], a smooth, positive definite and radially un-bounded function V c is called a Control Lyapunov Function (CLF) for system(22) if for all x = 0,

V cxB = 0 =⇒ V c

xf < 0 (25)


Note that if V c is chosen as a CLF, it guarantees the stability of thecontroller (24), since

V c = V cxf + V c

xBu

= V cxf − V c

xBV c

xf +√[V c

xf ]2 + q(x)[V cxBB

TV cTx ]

V cxBB

TV cTx

BTV cTx

=−√

[V cxf ]2 + q(x)[V c

xBBTV cT

x ] < 0 V cxB = 0

V cxf < 0 V c

xB = 0

Also, by virtue of CLF, we can prove (see Appendix)

limV c

x B→0uop =

√q(x),

which means uop is bounded when V cxB goes to zero.

In most VSC design, a positive definite function V = 12σT σ is chosen to

facilitate the derivation of the switching control law. In particular in our casethe nonlinear optimal sliding mode is σ = σ1(x1) + ϕ(x1,x2, t). Because

VxB = [∂( 1

2σT σ)∂x

]TB

= σT [∂σ

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2][0B2

]

= σT ∂ϕ

∂x2B2,

Vxf = σT [∂σ1

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2][

f1 +G1ϕf2

].

and ∂ϕ∂x2

B2 is full rank, we have

VxB = 0 =⇒ σ = 0 =⇒ Vxf = 0.

Thus, V is not a CLF here.In order to apply the Sontag’s formula, it is necessary to construct a CLF.

The following Theorem provides such a CLF which incorporates the σT σ asits subset.

Theorem 3. The positive definite function

V c = V1(x1) +12σT σ (26)

is a control Lyapunov function for system (22), where σ is defined in (11)and

V1 = V ∗,

V ∗ is the C1 positive semidefinite solution of the HJB equation (9).


Proof. It is easy to obtain

V cxB = V T

1xB + [∂( 1

2σT σ)∂x

]TB

= 0 + σT [∂σ

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2][0B2

]

= σT ∂ϕ

∂x2B2. (27)

On the other hand,

V cxf = [

∂V1

∂x]T

[f1 +G1ϕ

f2

]+ [

∂( 12σT σ)∂x

]T[

f1 +G1ϕf2

]

= [∂V1

∂x1]T [f1 +G1ϕ ] + σT [

∂σ1

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2][

f1 +G1ϕf2

]

Since ∂ϕ∂x2

B2 is full rank, from (11) and (27) we have

V cxB = 0 =⇒ σ = 0 =⇒ ϕ = −σ1(x1). = −1

2R−1GT

1 V∗Tx1

.

Using the relationship (9),

V cxf = [

∂V1

∂x1]T [f1 +G1ϕ ]

= −l(x1) +14[∂V1

∂x1]TG1R

−1GT1 [∂V1

∂x1] + [

∂V1

∂x1]TG1ϕ

= −l(x1) +14[∂V1

∂x1]TG1R

−1GT1 [∂V1

∂x1]− 1

2[∂V1

∂x1]TG1R

−1GT1 [∂V1

∂x1]

= −l(x1)− 14[∂V1

∂x1]TG1R

−1GT1 [∂V1

∂x1].

< 0

Hence V C is a CLF.

Remark 5. To apply the suboptimal controller (24), the system model shouldbe known a priori. It will lose the effectiveness when the system perturbationbecomes dominant. An effective way to overcome this limitation is to integrateoptimal control approaches with VSC to gain the robustness.

5 Quasi-optimal Variable Structure Control

Now we are in a position to give the main result as summarized in the The-orem below. A new quasi-optimal VSC scheme is constructed through in-corporating the nonlinear optimal sliding mode, suboptimal control with theproposed CLF, and a switching control,


Theorem 4. The integrated quasi-optimal VSC law

u = uop + uvsc

= −V cxf +

√[V c

xf ]2 + q(x)[V cxBB

TV cTx ]

V cxBB

TV cTx

BTV cTx

− [ρ(x, t) + δ]γ(1− εb)‖γ‖ (28)

with

γT = σT ∂ϕ

∂x2B2

ρ(x, t) = εb‖uop‖+ ρm(x, t). (29)

achieves asymptotic stability for system (1), where V c is a CLF defined in(26).

Proof. Choose the CLF given in (26) and differentiate it,

V c = [∂V1

∂x]T f + [

∂V1

∂x]T

[0

B2[(I +∆B2)u + d]

]

+σT [∂σ

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2]f +

[0

B2(I +∆B2)

]u +

[0

B2d

]

= [∂V1

∂x]T f + σT [

∂σ

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2]f +

[0

B2d

]+

[0

B2∆B2

]uop

+[

0B2(I +∆B2)

]uvsc+ σT [

∂σ

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2]Buop.

For

uop = −V c

xf +√[V c

xf ]2 + q(x)[V cxBB

TV cTx ]

V cxBB

TV cTx

BTV cTx

and

V cxB = [

∂V1

∂x]T

[0B2

]+ σT [

∂σ

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2]B

= σT [∂σ

∂x1+

∂ϕ

∂x1

∂ϕ

∂x2]B,

we have

V c = [∂V c

∂x]T f + σT ∂ϕ

∂x2B2[d +∆B2uop + (I +∆B2)uvsc]

−σT [∂σ

∂x1+

∂ϕ

∂x2

∂ϕ

∂x2]B

·Vcxf +

√[V c

xf ]2 + q(x)[V cxBB

TV cTx ]

σT [ ∂σ∂x1

+ ∂ϕ∂x1

∂ϕ∂x2

]BBTV cTx

BTV cTx


= σT ∂ϕ

∂x2B2[d +∆B2uop + (I +∆B2)uvsc]−

√ξ

= γT (d +∆B2uop) + γT uvsc + γT∆B2uvsc −√ξ

≤ ‖∆B2uop + d‖ · ‖γ‖+ γT uvsc + γT∆B2uvsc −√ξ, (30)

where ξ = [V cxf ]2 + q(x)[V c

xBBTV cT

x ].Note that ρ(x, t) = εb‖uop‖+ρm(x, t) ≥ ‖∆B2uop+d‖. Similar to Section

2, choose

uvsc = − [ρ(x, t) + δ]γ(1− εb)‖γ‖ ,

where δ ≥ 0 is a constant and ρ is defined in (29). Substituting uvsc into(30), we have

V c ≤ ρ(x, t)‖γ‖ − γT [ρ(x, t) + δ]γ(1− εb)‖γ‖ + εb

‖γT ‖[ρ(x, t) + δ]‖γ‖(1− εb)‖γ‖ −

√ξ

= −δ‖γ‖ −√ξ < 0. (31)

Remark 6. The proposed controller (28) is synthesized by uop and uvsc whichare made to function in a complementary manner to fulfill the control ob-jective: while in the region dominated by the nominal part, the suboptimalcontrol uop will govern the system response and drive the system state ap-proaching the equilibrium in a suboptimal fashion; while in the disturbancedominant region, the VSC part uvsc will take over the control task to warrantthe robustness.

Remark 7. From Theorem 4, the advantage of the quasi-optimal VSC is im-mediately obvious. While the control law (28) in general guarantees the stabil-ity and the reaching condition (31), we now have one extra degree of freedomin tuning the controller, in the sequel tuning control performance. In par-ticular, we can choose an appropriate quantity q(x) in (23) to change theoptimal control part. A large q(x) will lead to time-optimal control as thesystem state becomes the main concern. On the contrary, by reduction ofq(x), the control input becomes the main concern (more penalized), and onecan expect less control effort and smoother responses.

Remark 8. : The switching gains of the typical VSC (6) and the quasi-optimalVSC (28) are [εb‖uc‖ + ρm(x, t)]/(1 − εb) and [εb‖uop‖ + ρm(x, t)]/(1 − εb)respectively. Note that uop is bounded with

√q(x) and will approach zero

when the system state approaches equilibrium, whereas uc may still be largeif f2(x1,x2, t) is nonvanishing around the equilibrium. Therefore the quasi-optimal VSC may have much lower switching gain than that of the typicalVSC near the equilibrium.


6 Illustrative Examples

We present two examples in this section. The first shows how the proposedquasi-optimal VSC method can be successfully applied to a highly nonlinearsystem. The second example clearly shows that the nonlinear optimal slidingmode cannot be approximated by a locally linearized one.

6.1 Attitude Control of A Rigid Spacecraft

Consider the following attitude motion model of a rigid spacecraft [8],

ρ = H(ρ)ωω = J−1S(ω)Jω + J−1(u + do)

(32)

where ρ ∈ R3 is the Cayley-Rodrigues parametric vector describing the bodyorientation, ω ∈ R3 is the angular velocity vector in a body-fixed frame,u ∈ R3 is the acting control torque, S(ω) is a 3× 3 skew-symmetric matrix,that is

S(ω) =

0 ω3 −ω2

−ω3 0 ω1

ω2 −ω1 0

,

the matrix-valued function H: R3 → R3×3 denotes the kinematics Jacobianmatrix for the Cayley-Rodrigues parameters, given by

H(ρ) =12(I − S(ρ) + ρρT ),

where J = ∆J−1J0, J0 > 0 is the nominal part of the inertia matrix, here itis assumed J0 = diag(10, 15, 20)kgm, and

∆J = diag(∆j1,∆j2,∆j3),do = [do1, do2, do3]T ,

where perturbations ∆ji and doi(i = 1, 2, 3) are as follows

∆ji = 1 + 0.1 sin(it),doi = sin(10t).

The rigid spacecraft model (32) can be converted to our standard formatas follows,

x1 = G1x2

x2 = f2 +B2[I +∆B2)]u + d (33)


where

x1 = ρ,

x2 = ω

G1 = H(x1),f2 = J−1

0 S(x2)J0x2

B2 = J−10 ,

∆B2 =

0.1 sin(t) 0 0

0 0.1 sin(2t) 00 0 0.1 sin(3t)

,

d = [∆JS(x2)∆J−1 − S(x2)]J0x2 +∆Jd0

To design the nonlinear optimal sliding mode, we need to solve the HJBequation for the x1-subdynamics. Unfortunately, it is not a feasible task. Thuswe will construct an inverse optimal sliding mode by applying the methodillustrated in Section 3.3.

Rewrite the x1-subdynamics of the model (33) as

x1 = G1(x1)v. (34)

Finding a stabilizing feedback control law which has the form as defined in(16) is a crucial step in the inverse optimal sliding mode construction. Sucha control law is chosen to be

v = −R−1G1(x1)TV Tx1

(35)

where R = RT > 0 and V = 12xT

1 x1 is a positive definite and radiallyunbounded Lyapunov function. Note that

V = [∂V

∂x1]T x1

= xT1 x1

= xT1 G1(−R−1G1

TV Tx1)

= −xT1 G1R

−1G1T x1

< 0, (36)

for all x1 = 0. Thus the control law (35) stabilizes the system (34).According to Section 3.3, the optimal control law

v∗ = 2v = −2R−1G1T x1

minimizes the cost

J =∫ ∞

0

[l(x1) + vTRv]dt (37)

where

l(x1) = 4xT1 G1R

−1G1T x1


and the optimal value function is

V ∗ = 2xT1 x1.

Thus the corresponding inverse optimal sliding mode is

σ = σ1 + x2 = 2R−1G1T x1 + x2 (38)

Now let us construct the suboptimal control in terms of the system nominalpart. According to Theorem 4 in Section 5, the CLF for the nominal part ofmodel (32) is

V c = V1 +12σT σ

= V ∗ +12(σ1 + x2)T (σ1 + x2)

= 2xT1 x1 +

12(2R−1G1

T x1 + x2)T (2R−1G1T x1 + x2).

Thus, the suboptimal control part of the proposed controller is

uop = −V cxf +

√[V c

xf ]2 + q(x)[V cxBB

TV cTx ]

V cxBB

TV cTx

BTV cTx (39)

where

x = [xT1 , xT

2 ]T (40)

f = [G1T , fT

2 ]T

B = [0, BT2 ]

T

and q(x) is a positive definite function.It is straightforward to determine the switching control part uvsc of the

proposed controller

uvsc = − (ρ+ δ)γ(1− εb)‖γ‖ (41)

where δ > 0 is a constant,

εb = 0.1γT = σTB2 = (2R−1G1

T x1 + x2)TB2

ρ = εb‖uop‖+ ρm

ρm =10√29

11[max(x21,x22) + max(x21,x23) + max(x22,x23)]‖x2‖

+3.3.

The quasi-optimal VSC for the rigid spacecraft attitude motion model (32)is

u = uop + uvsc (42)


with uop defined in (39) and uvsc defined in (41).Figure 1 shows the x11 − x21 phase planes of the suboptimal controller

(39) and the proposed controller (42) respectively. The closeness of the twocurves indicates that the quasi-optimal VSC does retains the suboptimalcontrol performance where the system nominal part is predominant.

Figure 2 shows the system responses near the equilibrium under the sub-optimal controller (39) and the proposed controller (42). We observe thatthe system state, when near the equilibrium, cannot converge anymore withthe suboptimal controller alone. On the contrary, the system state keeps con-vergent with the proposed controller, owing to the robustness property ofVSC.

Figure 3 and Fig. 7 show the possibility of adjusting system responses bychanging two weightings q(x) in the suboptimal control part and R in theinverse optimal sliding mode (38), respectively. In Fig. 3, q(x) in the proposedcontroller (42) is chosen to be a(xT

1 x1 + xT2 x2). The solid line, dashed line

and dashdotted line in Fig. 3 show the values of the switching quantity σ1

(the first element of σ) with a = 10000, 100, 1, respectively. We observe thata larger a, i.e. a large q(x), will expedite the system response and drive thesystem state to reach the sliding mode faster. Figure 4, Fig. 5 and Fig. 6 showthe corresponding control profiles. Next let us check R, which according to(37) should have effect on the convergence of the states x1 when the system isin the sliding mode. For simplicity we only show the responses of the first statex1. In Fig. 7, the solid line is obtained with R = diag(0.1, 1, 1) and the dashedline is obtained with R = diag(0.5, 1, 1). We can observe that a smaller R willexpedite the system response. These results clearly illustrate that the newquasi-optimal VSC possesses both robust and optimal properties. Figure 8shows the control profiles. Note that the control gain is very large, this isbecause we use a very large q(x), i.e. 106(xT

1 x1+xT2 x2), in order to force the

system dynamics to reach the inverse optimal sliding mode in a short time.

6.2 Inverse Optimal Sliding Mode vs. Linear Sliding Mode

Here we show that a nonlinear sliding mode may be indispensable when thesystem is nonlinear in nature. Consider the following x1-subdynamics

x1 = f1(x1) +G1(x1)x2, (43)

where x1 = [x11, x12]T , x2 = [x21, x22]T and

f1 =[−x11 sin2(x11)− x11

−x12 sin2(x12)− x12

]

G1 =[1 + cos2 x11 − x2

11 00 1 + cos2 x12 − x2

12

].

A globally stable inverse optimal sliding manifold

σ = 2R−1GT1 x1 + x2. (44)


for system (43) can be achieved by applying the method given in Sec. 3.3.For comparison purposes, a linear optimal sliding mode is also designed

based on the linearized model at x1 = 0. The linearized model is obtainedby removing those higher order infinifestimal terms

x1 = A11x1 +A12x2, (45)

where A11 = −I and A12 = 2I. Consider the optimal control task below

x1 = A11x1 +A12v

infv(·)

∫ ∞

0

[xT1 Qx1 + vTRv]dt.

where Q = QT > 0 and R = RT > 0. The algebraic Ricatti equation for thisoptimal control problem is

PA11 +AT11P − PA12R

−1AT12P +Q = 0

or − 2P − 4PR−1P +Q = 0, (46)

and the optimal control law is given by

v∗(x1) = −2R−1Px1,

where P is the solution of the algebraic Ricatti equation (46). Then theoptimal sliding mode defined in (14) is as follows

σ = 2R−1Px1 + x2. (47)

Figure 9 shows the responses of x11 when the system is in the linear slidingmode (47) with different initial values x1(0) = [0.5, 0.4]T and x(0) = [5, 4],Q = R = I. Fig. 10 shows that the system with larger initial values x(0) =[5, 4] is unstable! The linear-type sliding mode applied to the nonlinear x1-subdynamics results in an unstable sliding manifold. Figure 11 shows theresponses of x11 when the system is in the nonlinear sliding mode with thesmaller initial values x1(0) = [0.5, 0.4]T . Fig. 12 shows the system responseswith the larger initial values x(0) = [5, 4]. Solid lines and dashdotted linesare obtained with R = I and R = diag(0.01, 1, 1) respectively. We observethat the nonlinear sliding mode with inverse optimality ensures the globalstability of the sliding manifold, and meanwhile retains the optimal controlproperties, such as the possibility of manipulating responses through tuningthe weightings R.

7 Conclusion

In this paper, we proposed a new control strategy - quasi-optimal VSC, byintegrating VSC with optimal sliding mode as well as nonlinear suboptimalcontrol based on CLF. The nonlinear optimal sliding mode design provides a


systematic way to guarantee a stable nonlinear sliding manifold, consequentlymaking VSC approaches applicable to more general classes of nonlinear sys-tems. In the new control method, the suboptimal control and VSC are madeto function in a complementary manner. It performs as the optimal controlwhen the system is dominated by the nominal part, and as the typical VSCwhen the system perturbations become dominant. Both nonlinear optimalsliding mode design and suboptimal controller design offer extra degrees offreedom in the weight selection so as to meet the different performance re-quirements.

References

1. Utkin, V. I. (1992) Sliding Modes in Control Optimization. Springer-Verlag,Berlin

2. Young, K. D. (1997) Sliding-mode Design for Robust Linear Optimal Control.Automatica, 33, 1313-1323

3. Yu, X. and Man, Z. (1998) Multi-input Uncertain Linear Systems with TerminalSliding-mode Control. Automatica, 34, 389-392

4. Young, K. D., Utkin, V. I. and Ozguner, U. (1999) A Control Engineer’s Guideto Sliding Mode Control. IEEE Transactions on Control Systems Technology,7, 328-342

5. Chien, C. J. and Fu, L. C. (1999) Adaptive Variable Structure Control. Newnes,Oxford

6. Bartolini, G., Ferrara, A. and Stotsky, A. (1999) Robustness and Performanceof an Indirect Adaptive Control Scheme in Presence of Bounded Disturbances.IEEE Transactions on Automatic Control, 44, 789-793

7. Edwards, C., Spurgeon, S. K. and Patton, R. J. (2000) Sliding Mode Observersfor Fauly Detection and Isolation. Automatica, 36, 541-553

8. Krstic, M. and Tsiotras, P. (1999) Inverse Optimal Stabilization of a RigidSpacecraft. IEEE Transactions on Automatic Control, 44, 1042-1049

9. Sepulchre, R., Jankovic, M. and Kokotovic, P.V. (1997) Constructive NonlinearControl. Springer, London, New York

10. Primbs, J. A., Nevistic, V. and Doyle, J. C. (1999) Nonlinear Optimal Control:A control Lyapunov Function and Receding Horizon Perspective. Asian Journalof Control, 1, 14-24

Appendix

Define

θ = Vxf ,

wT = VxB.

Since V (x) is a Control Lyapunov Function, whenever x = 0, we have

w = 0 ⇒ θ < 0.


Assume that at a point x0, V cxB|x=x0 = 0 and V c

xf |x=x0 < 0. Since both fand B are continuous with respect to all arguments, and V c is continuouslydifferentiable, ∀ε > 0, ∃δ > 0, such that when ‖x − x0‖ < δ, there are‖V c

xB‖ < ε and V cf < 0.Thus, when ‖x − x0‖ < δ, we have

‖uop‖ =

∥∥∥∥∥∥−V c

xf +√[V c

xf ]2 + q(x)[V cxBB

TV cTx ]

V cxBB

TV cTx

BTV cTx

∥∥∥∥∥∥=

∣∣∣∣∣θ +

√θ2 + q(x)wT w

wT w

∣∣∣∣∣ · ‖w‖

≤ −|θ|+ |θ|+ √q(x)‖w‖

‖w‖2‖w‖

=√q(x).

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1Proposed Method Suboptimal Control

Fig. 1. ρ1 − ω1 phase plane under suboptimal control and quasi-optimal VSC


10 12 14 16 18 20 22 24 26 28 30−1

−0.5

0

0.5

1

1.5

2

2.5

3x 10

−3

t

p1

proposed methodoptimal control

Fig. 2. ρ1 under suboptimal control and quasi-optimal VSC (near equilibrium)

0 1 2 3 4 5 6 7 8 9 10−5

0

5

10

15

20

25

30

t

Sw

itchi

ng S

urfa

ce

a = 10000a = 100 a = 1

Fig. 3. Switching surface σ1 under quasi-optimal VSC with different q(x)

−1 0 1 2 3 4 5 6 7 8 9 10−300

−250

−200

−150

−100

−50

0

50

100

t

u1

Fig. 4. Control profile of quasi-optimal VSC with q(x) = 10000(xT1 x1 + xT

2 x2)


−1 0 1 2 3 4 5 6 7 8 9 10−30

−25

−20

−15

−10

−5

0

5

10

t

u1


2 x2)

−1 0 1 2 3 4 5 6 7 8 9 10−6

−4

−2

0

2

4

6

t

u1


2 x2)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

p1

R = diag(0.1, 1, 1)R = diag(0.5, 1, 1)

Fig. 7. System responses in inverse optimal sliding mode with different R


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−8000

−6000

−4000

−2000

0

2000

4000

6000

t

u1

R = diag(0.1, 1, 1) R = diag(0.5, 1, 1)

Fig. 8. Control Profiles of quasi-optimal VSC with different R

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

X11

R = diag(1,1) R = diag(0.01,1)

Fig. 9. System responses with a local optimal sliding mode, x(0) = [0.5, 0.4]

0 0.01 0.02 0.03 0.04 0.05 0.060

10

20

30

40

50

60

70

80

90

100

t

X11

Fig. 10. System responses with a local optimal sliding mode, x(0) = [5, 4]


0 0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

X11

R = (1,1) R = (0.1,1)

Fig. 11. System responses with inverse optimal sliding mode, x(0) = [0.5, 0.4]

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

X11

R = diag(1,1) R = diag(0.1,1)

Fig. 12. System responses with inverse optimal sliding mode, x(0) = [5, 4]


Robust Control of Infinite-DimensionalSystems via Sliding Modes

Yuri Orlov

CICESE Mexican Research Center, P.O.Box 434944, San Diego CA 92143, USA

Abstract. Infinite-dimensional control systems, driven by a discontinuous feed-back, are under study. Discontinuous control algotithms are developed. The algo-rithms ensure desired dynamic properties as well as robustness of the closed-loopsystem against matched disturbances. The theory presented is illustrated by appli-cations to heat processes and mechanical distributed oscillators.

1 Introduction

Many important plants, such as time-delay systems, flexible manipulators andstructures as well as heat transfer processes, combustion, and fluid mechanicalsystems, are governed by functional and partial differential equations or, moregeneral, equations in a Hilbert space. As these systems are often describedwith a significant degree of uncertainty, robust controller design for thesesystems presents a challenging problem.

Sliding mode control of finite-dimensional systems is known to guaran-tee a certain degree of robustness with respect to unmodal dynamics. Sincethe sliding mode equation is control-independent, the approach based on thedeliberate introduction of sliding motions into the control system splits thecontrol problem into two independent problems of lower dimensions. First, adiscontinuity manifold with the prescribed dynamic properties of the slidingmotion is designed and then a discontinuous control, which ensures the slidingmotion on this manifold, is synthesized. Apart from decoupling the originalcontrol problem, the sliding mode approach makes the closed-loop systeminsensitive to matched disturbances. Due to these advantages and simplicityof implementation, sliding mode controllers are widely used in various appli-cations. An overview of finite-dimensional sliding mode control theory andapplications can be found in [20].

The first few papers [1,12,16] on the application of sliding mode control al-gorithms to DPS corroborated their utility for infinite-dimensional systems aswell and motivated further theoretical investigations [17,23], which were con-fined, however, to semilinear parabolic systems with a finite horizon. In orderto describe the sliding modes in these systems, the sliding mode equation wasshown in [17] to be well-posed via relating the discontinuous control law tothe continuous one. The conditions for the infinite-dimensional sliding modeto exist were obtained in [23] through a finite-dimensional Faedo-Galerkinapproximation of the original discontinuous control system.


Subsequently, a set of sliding mode control algorithms has been proposedfor distributed parameter plants governed by uncertain partial differentialequations (cf. [21,22] and the references quoted therein). All algorithms fol-lowed the conventional finite-dimensional approach, which implies that eachcomponent of a control action undergoes discontinuities on its own surfaceand as a result a sliding mode is enforced in their intersection.

This approach, however, becomes invalid in the general infinite-dimensionalsetting because neither control input nor sliding manifold is representablein a component-wise form. Thus, infinite-dimensional discontinuous controlmethods developed in the present chapter make a step beyond the finite-dimensional treatment.

The chapter is outlined as follows. In Section 2 we demonstrate someattractive features of discontinuous control systems in a Hilbert space andmotivate the subsequent theoretical development. We present here an infinite-dimensional system driven by discontinuous control signals along the discon-tinuity manifold on an infinite time interval. The discontinuous control lawresults from the Lyapunov min-max approach, the origins of which may befound in [7,8]. An extension of this approach to infinite-dimensional systemscan be found in [15,18]. Based on the extension, the control is synthesizedto guarantee that the time-derivative of a Lyapunov function, selected for anominal, exponentially stable system, is negative on the trajectories of thesystem with perturbations caused by uncertainties of a plant operator and en-vironment conditions. The approach gives rise to the control action, referredto as a unit control, the norm of which is equal to 1 everywhere with theexception of the discontinuity manifold. The closed-loop system enforced bythe unit control is shown to be exponentially stable and robust with respectto matched disturbances.

Section 3 presents the decomposition-based synthesis of a discontinuouscontrol law, which imposes the desired dynamic properties as well as robust-ness with respect to matched disturbances on the closed-loop system. If theundisturbed motion of the system contains two components, one of them sta-ble and another one belonging to a finite-dimensional subspace, the controlsynthesis, as is shown in Section 4, is split into two independent synthesisprocedures. The first procedure uses the standard finite-dimensional settingwhile the second one is carried out within the infinite-dimensional subspace ofthe exponentially stable internal dynamics. The latter procedure is developedin Section 2 of the present chapter.

As an illustration of the capabilities of the above procedure, a scalarunit controller for a minimum phase system of a finite relative degree isconstructed in Section 5.

202 Y. Orlov

2 Unit Feedback Control Synthesis

In this section we present a synthesis of discontinuous unit feedback controlsignals and discuss some attractive capabilities of discontinuous controllersin addressing both finite and infinite-dimensional systems.

2.1 Synthesis in Eucleadean State Space

We start our investigation with an affine system

x = f(x, t) + B(x, t)u + h(x, t) (1)

with finite-dimensional state and control vectors x ∈ Rn, u ∈ Rm and state-dependent n-vectors f(x, t) and h(x, t), and matrix B(x, t) ∈ Rn×m. Thevector h(x, t) represents the system uncertainty and its influence on the con-trol process should be rejected.

The equation

x = f(x, t) (2)

represents an open loop nominal system which is assumed to be asymptoti-cally stable with some apriori known Lyapunov function:

V (x) > 0, Wo =dV

dt|h=0,u=0= grad(V )T f < 0. (3)

The perturbation vector h(x, t) is assumed to satisfy the matching condition[4]

h(x, t) ∈ span(B).

In other words, it is assumed that there exists vector λ(x, t) ∈ Rm such that

h(x, t) = B(x, t)λ(x, t). (4)

Here λ(x, t) may be an unknown vector with an apriori known upper scalarestimate λo(x, t), i.e.,

‖ λ(x, t) ‖< λo(x, t). (5)

The time derivative of V (x) on the trajectories of the perturbed system (1),(4)is of the form

W =dV

dt= Wo + grad(V )TB(u + λ) < 0. (6)

For the control

u = −ρ(x, t)BT grad(V )

‖ BT grad(V ) ‖ , (7)

203Robust Control of Infinite-Dimensional Systems via Sliding Modes

depending on the upper estimate of the unknown disturbance, with a scalarfunction

ρ(x, t) > λo(x, t)

and

‖ grad(V ) ‖2= [grad(V )TB][BT grad(V )]

the time derivative of the Lyapunov function V (x)

W = Wo − ρ(x, t) ‖ grad(V ) ‖ +grad(V )TBλ(x, t) <

Wo− ‖ BT grad(V ) ‖ [ρ(x, t) − λo(x, t)] < 0

is negative. This means that the perturbed system with control (7) is asymp-totically stable, too.

Two important features should be underlined for the system with control(7):

1. The control signal is a discontinuous function of the system state and itundergoes discontinuities in the (n−m)-th dimensional manifold

s(x) = BT grad(V ) = 0. (8)

2. The disturbance h(x, t) is rejected due to enforcing the sliding mode inthe manifold s(x) = 0.

Note that the equivalent value ueq of the control signal (7) is equal to −λ(x, t)along the discontinuity manifold s(x) = BT grad(V ) = 0, which is not,generally speaking, the case for the control law (7) beyond the manifold,s(x) = BT grad(V ) = 0. Meanwhile, beyond this manifold the norm

‖ BT grad(V )‖ BT grad(V ) ‖‖

of the control signal (7) with the gain ρ(x, t) = 1 is equal to 1 for any valueof the state vector. This explains the term unit control for the control signal(7).

It is of interest to note that in contrast to the conventional sliding modecontrol signals which undergo discontinuities whenever a component of thesliding manifold changes sign, the unit control action is a continuous statefunction until the manifold s(x) = 0 is reached. Due to this difference the unitcontrol method turns out to be an appropriate tool to design a discontinuousinfinite-dimensional system with control inputs which are not (or even cannot be) represented in a component-wise form.

204 Y. Orlov

2.2 Synthesis in Hilbert State Space

In order to illustrate the fact that discontinuous infinite-dimensional systemscan also be driven along discontinuity manifolds, we consider a dynamicalsystem

x = e(x), x(0) = x0 ∈ H (9)

in a real Hilbert space H enforced by the unit control action e(x) = −x/‖x‖which undergoes discontinuities in the trivial manifold x = 0. Since the norm‖x‖ =

√(x, x) in the Hilbert space is defined via its inner product (·, ·), then

12d‖x‖2

dt= (x(t), x(t)) = −‖x(t)‖,

and therefore ‖x(t)‖ = (‖x0‖ − t) for t ≤ ‖x0‖. Hence, in the infinite-dimensional system (9) starting from the time moment t = ‖x0‖, thereappears a sliding mode in the discontinuity manifold x = 0. Clearly, thesliding mode is unambiguously set by the manifold equation of x = 0 regard-less of uniformly bounded additive dynamic nonidealities h(x, t) such that‖h(x, t)‖H < 1 for all t ≥ 0, x ∈ H, which are rejected by the unit control.In this case the sign of the time derivative of the Lyapunov functional alongthe trajectories of the perturbed system x = e + h remains negative. How-ever, in general, neither the unit control belongs to the state space nor thediscontinuity manifold is trivial, so that their synthesis presents a formidableproblem.

According to the unit feedback approach, developed in the sequel, a lin-ear discontinuity manifold cx = 0 in the control space U , which differs fromthe state space H, may be constructed in compliance with some perfor-mance criterion, particularly, according to the Lyapunov min-max approach,whereas a sliding mode in the manifold is enforced by the corresponding unitcontrol M(x, t)e(cx) = −M(x, t)cx/‖cx‖U , possibly with a non-unit gainM(x, t) = 1. This design idea is now illustrated for an uncertain dynamicsystem governed by a differential equation

x = Ax + f(x, t) + bu(x, t), x(0) = x0 ∈ D(A) (10)

where the state x(t) and control signal u(x, t) are abstract functions with val-ues in Hilbert spaces H and U , respectively; A is the infinitesimal generatorof an exponentially stable semigroup TA(t) on H; b ∈ L(U,H). The oper-ator function f(x, t) with values in H represents the system uncertainties,whose influence on the control process should be rejected. Throughout thepresent chapter, this function is assumed to be continuously differentiable inall arguments and satisfy the matching condition

f(x, t) = bh(x, t) (11)


where the uncertain function h(x, t) has an a priori known upper scalar esti-mate N(x) ∈ C1, i.e.

‖h(x, t)‖U < N(x) for all x ∈ H, t ≥ 0. (12)

In order to apply the afore-mentioned Lyapunov approach to the infinite-dimensional system (10), let us note that the positive definite solution WA =∫ ∞0

T ∗A(t)TA(t)dt to the Lyapunov equation WAA + A∗WA = −I with the

identity operator I assigns the quadratic Lyapunov functional

V (x) = (WAx, x)

for the nominal system x = Ax. Then, taking into account (11) and differ-entiating the Lyapunov functional with respect to t along the trajectories ofthe perturbed system (10), we obtain

dV/dt = (WAx(t), x(t)) + (WAx(t), x(t)) = −(x, x) +2(WAx(t), b(u + h)) = −(x, x) + 2(b∗WAx(t), u + h). (13)

A straightforward application of the Lyapunov min-max approach, which re-quires minimization of the right-hand side of (13) under the control constraint‖u(·)‖U ≤ M = const, results in the unit control

u(x) = −Me(b∗WAx) = −M b∗WAx

‖b∗WAx‖U . (14)

Given the state-dependent gain M = N(x), the time derivative of the Lya-punov functional along the trajectories of the perturbed system (10) drivenby the unit control (14) is forced to be negative, namely

dV/dt ≤ −(x, x) ≤ − 1‖WA‖ (WAx, x) = − 1

‖WA‖V (x) (15)

for all x ∈ H (including the discontinuity manifold!), regardless of admissibleplant perturbations f(x, t). This guarantees the exponential stability of theclosed-loop system. Thus, the unit control (14) with the gain M = N(x)rejects any admissible perturbation f(x, t) and imposes the desired dynamicand robustness properties on the uncertain system (10). Along with the rela-tive simplicity of the implementation of unit control signals (cf. that of [5,10]),these properties make the use of unit controllers in the infinite-dimensionalcase attractive.

3 Decomposition of Synthesis Procedure

3.1 Decomposition in Eucleadean State Space

According to the finite-dimensional treatment, proposed in [3,19,20] for theaffine system (1), the unit control synthesis procedure consists of two steps.

206 Y. Orlov

First, a sliding mode is selected to have the prescribed dynamic propertiesof the motion in the sliding mode by a proper choice of the discontinuitymanifold s = 0. And second, a discontinuous control is constructed to guar-antee existence of the sliding motion along this manifold. Once the manifolds(x) = 0 has been selected in compliance with some performance criterion,the control input is designed in the form (7):

u = −ρ(x, t)GT s(x)

‖ GT s(x) ‖ , (16)

with G = ∂s∂xB, G is assumed to be nonsingular. The equation of a motion

projection of the system (1) on the subspace s is of the form

s = ∂s∂x

(f + h) + Gu. (17)

The conditions for the trajectories to converge to the manifold s(x) = 0and the sliding mode to exists in this manifold may be derived based on theLyapunov function

V =12sT s > 0 (18)

with the time derivative

V = sT ∂s∂x

(f + h) − ρ(x, t) ‖ GT s(x) ‖<

‖ GT s(x) ‖ ·[‖ G−1 ∂s∂x

(f + h) ‖ −ρ(x, t)]. (19)

For ρ(x, t) >‖ G−1 ∂s∂x(f +h) ‖ the value of V is negative and therefore the

state will reach the manifold s(x) = 0 in a finite time interval for any initialconditions and then the sliding mode with the desired dynamics will occur.The boundedness of the interval preceding the sliding motion follows fromthe inequality resulting from (18),(19):

V < −γV 1/2, γ = const > 0

with the solution

V (t) < (−γ

2t +

√Vo)2, Vo = V (0).

Since the solution vanishes after some ts <2γ

√Vo, the vector s vanishes so

that the sliding mode starts after a finite time interval.In order to describe the sliding mode one should substitute the equivalent

control value

ueq = −G−1 ∂s∂x

(f + h),


i.e. the continuous solution of the equation s = 0 with respect to u, into (1)for u. Due to the equivalent control method [20] the resulting equation

x = f + h−G−1 ∂s∂x

(f + h), (20)

referred to as a sliding mode equation, governs the system motion alongthe sliding manifold s = 0. Since the sliding mode equation is control andmatched disturbance-independent (indeed, equation (20) subject to (4) takesthe form x = f − G−1 ∂s

∂xf ), this approach leads to decomposition of theoriginal design problem into two independent problems and permits construc-tion of a control system which is insensitive to matched disturbances.

3.2 Decomposition in Hilbert State Space

It is clear, the afore-given procedure can also be used in the infinite dimen-sional system (10) with an infinitesimal operator A which generates a stronglycontinuous semigroup TA(t) rather than an exponentially stable semigroup,while the unit feedback disturbance rejection used in Section 2 is no longer inforce. The control problem is then split into the selection of a proper Hilbertspace S and the linear discontinuity manifold

cx = 0, c ∈ L(H,S) (21)

with the desired zero dynamics

x1 = (A11 −GA21)x1, (22)

and design of a unit feedback controller, which ensures the motion of thesystem along this manifold. In order to derive the sliding mode equation (22)one should represent the Hilbert space

H = H1 ⊕H2.

via the kernel

H1 = ker c = x1 ∈ H : cx1 = 0 ⊆ H

of the operator c and its complementary H2 ⊆ H (see, e.g., [11]), and thenrewrite equation (10) in terms of variables x1(t) ∈ H1 and x2(t) ∈ H2 :

x1 = A11x1 + A12x2 + P1f(x1, x2, t) + P1bu(x1, x2, t), t ≥ 0,x1(0) = x0

1, (23)x2 = A21x1 + A22x2 + P2f(x1, x2, t) + P2bu(x1, x2, t), t ≥ 0,

x2(0) = x02. (24)

Here x1(t) ⊕ x2(t) = x(t), x01 ⊕ x0

2 = x0, Pi is the projector on the subspaceHi, Aij = PiAj is the operator from Hj to Hi, Aj = A|Hj

is the operator

208 Y. Orlov

restriction on Hj , i, j = 1, 2. Clearly, the discontinuity manifold (21), writtenthrough the new coordinates, takes the form x2 = 0.

Assuming that the operator P2b from U to H2 is boundedly invertible (i.e.,the operator (P2b)−1 from H2 to U is bounded) the sliding mode equation

x1 = Ax1 + [P1 −GP2]f(x1, 0, t) (25)

in the discontinuity manifold x2 = 0 is derived according to the equivalentcontrol method (the extension of the method to infinite-dimensional systemsis justified in [15]) by substituting the continuous solution

ueq(x, t) = −(P2b)−1[A21x1 + P2f(x1, 0, t)] (26)

of the equation x2 = 0 into (23) for u(x1, x2, t). Since the external disturbancesatisfies the matching condition (11), we obtain that [P1 − GP2]f(x, t) =[P1 − P1b(P2b)−1P2]bh(x, t) = 0 and hence the sliding mode equation (25)takes the disturbance-independent form (22).

Example 1: To exemplify the decomposition idea in the infinite-dimensionalsetting let us consider coupled thermal fields governed by the following partialdifferential equation

∂Q

∂t=

∂2Q

∂x2+ DQ + Fu(x, t),

0 < x < 1, t > 0,∂Q(0, t)/∂x = ∂Q(1, t)/∂x = 0, t ≥ 0,

Q(x, 0) = Q0(x), 0 ≤ x ≤ 1 (27)

where Q(x, t) ∈ Rn, u(x, t) ∈ Rm for all x ∈ R1, t ≥ 0; constant matricesD and F of appropriate dimensions are assumed to be controllable. Fromthe physical viewpoint, the problem consists of heating n similar plants byvirtue of m distributed sources; matrix D characterizes heat exchange withthe enviroment and between the plants.

Let the control signal drive system (27) to the manifold

S(Q) = cQ = c1Q1 + c2Q2 = 0, (28)

where

Q1 ∈ Rn−m, Q2 ∈ Rm, det c2 = 0, det (cF ) = 0.

Then the state equation can be represented in terms of Q1 and S as follows

∂Q1

∂t=

∂2Q1

∂x2+ D11Q1 + D12S + F1u(x, t), (29)

∂S

∂t=

∂2S

∂x2+ D21Q1 + D22S + cFu(x, t). (30)

Due to the equivalent control method, the system motion on manifold (28)is governed by the equation

∂Q1

∂t=

∂2Q1

∂x2+ RQ1, R = D11 − F1(cF )−1 (31)


that is obtained from substitution of the equivalent control value

ueq = −(cF )−1D21Q1,

resulting from (29), (30) with S(x, t) identically equal to zero. Applicabilityof the equivalent control method to the parabolic partial differential equation(27) is validated by Theorem 1 of [15].

Following the aforegiven design procedure, in the first step one needs tochoose a discontinuity manifold (28) to ensure the prescribed properties of themotion in the sliding mode. It is well-known that for the finite-dimensionalsystem

Q = DQ + Fu (32)

the equation of the sliding mode along the manifold cQ = 0 takes the formQ1 = RQ1 by virtue of the equivalent control technique. A matrix c for thecontrollable system may be chosen such that det (cF ) = 0 and the eigenvaluesof the matrix R take up the desired values with negative real parts ReλR <0. To specify the matrix c it suffices to represent system (32) in the canonicalform where the choice of the matrix becomes straightforward (see [20] fordetails). Based on this fact, the required rates of L2-convergence

limt→∞ ‖Q1(·, t)‖L2(0,1) = 0 (33)

of the state of the distributed parameter system (29) may be imposed as well.In order to obtain the desired rates of convergence (33), let us introduce theLyapunov functional

V (t) =∫ 1

0

QT1 (x, t)WQ1(x, t)dx (34)

where W =∫ ∞0

expRT t expRtdt is the positive definite solution of theLyapunov equation RTW + WR = −I, I is the identity matrix of an ap-propriate dimension, and find the time derivative of the functional along thetrajectories of (29):

V (t) = −2∫ 1

0

[∂Q1

∂x]T (x, t)W [

∂Q1

∂x]dx−

∫ 1

0

QT1 (x, t)Q1(x, t)dx. (35)

Denoting by λmax the maximal eigenvalue of the matrix W and bearing inmind that QT

1 WQ1 ≤ λmaxQT1 Q1, we arrive at

V (t) ≤ −λ−1maxV (t). (36)

By choosing eigenvalues of R with negative real parts sufficiently large inmagnitude, the value of λmax may be made as small as desired. Thus the ap-propriate choice of matrix c that assigns the desired allocation of eigenvalues

210 Y. Orlov

of R, ensures the Lyapunov functional convergence V (t) → 0 as t → ∞, aswell as (33) at desired rates.

In the second step, a discontinuous control is designed to drive the systemstate to the manifold S = 0. We demonstrate that the unit control

u(Q) = − M(Q)S1(Q)‖S1(Q(·, t))‖L2(0,1)

(37)

where

S1(Q) = (cF )−1S(Q), M(Q) = M0‖Q(·, t)‖L2(0,1),

M0 = ‖(cF )−1cD‖ + M1, M1 > 0 (38)

guarantees that in the closed-loop system (27), (37) starting from a finitetime moment there appears a sliding motion on the manifold S(Q) = 0 orequivalently, S1(Q) = 0. Indeed, differentiating the Lyapunov functional

V1(t) =∫ 1

0

ST1 (Q(x, t))S1(Q(x, t))dx

along the trajectories (27), employing integration by parts, applying bound-ary conditions and utilizing the control law (37), (38) yield

12V1(t) =

∫ 1

0

ST1 S1dx =

∫ 1

0

ST1 [∂2S1

∂x2+ (cF )−1cDQ + u]dx =

−∫ 1

0

[∂S1

∂x]T∂S1

∂xdx−

∫ 1

0

ST1 [

M(Q)S1(Q)‖S1(Q(·, t))‖L2(0,1)

−

(cF )−1cDQ]dx ≤ −M1

√V1(t). (39)

The solution to the latter inequality has been shown to vanish after thefinite time moment T = V1(0)/M1. Therefore, starting from T , the unitcontrol signal (37) enforces the system motion in the sliding mode along themanifold (28) and T → 0 as M1 → ∞. Thus the unit control approach leadsto a decoupling system design and ensures the desired rate of transient decay.

Further on we shall give general conditions which allow us to reduce theinfinite-dimensional control problem and use the well-known synthesis pro-cedures for finite-dimensional systems. Such a situation is shown to appearif the undisturbed motion of (10) under u(x, t) ≡ f(x, t) ≡ 0 contains twopartial components: one of them is stable and doesn’t require to be corrected,and another one belongs to a finite-dimensional subspace.


4 Disturbance Rejection in Exponentially StabilizableSystem

The aim of this section is to demonstrate how the uncertainties (11), (12) inthe exponentially stabilizable system (10) with a finite-dimensional unstablepart can be rejected by means of a unit stabilizing controller.

Throughout this section we assume that the pair A, b is exponentiallystabilizable and the spectrum σ(A) = σ1(A) + σ2(A) of the infinitesimaloperator A consists of two parts: one of them, σ1(A) = λ ∈ σ(A) : Re λ ≥ 0,is finite-dimensional and another one, σ2(A) = λ ∈ σ(A) : Re λ < 0, is inthe open left half-plane.

Let P1 and P2 be projectors corresponding to the spectral sets σ1(A), σ2(A),respectively, and Hj = PjH, j = 1, 2. Then it is well-known (see, e.g., [9,Section 1.5]) that

1. H = H1 ⊕H2, Hj are invariant with respect to A, i.e., AHj ⊂ Hj , j =1, 2;

2. the operator A1 = A|H1 is finite-dimensional, i.e., H1 = Rn;3. the operator A2 = A|H2 generates an exponentially stable semigroup

TA2(t) with some negative growth bound −β, i.e.,

‖TA2(t)‖ ≤ ωe−βt, ω > 0. (40)

If the operator A is compact resolvent then its spectrum σ(A) = λi∞i=1

would be discrete, and for any β > 0 there would exist a number l such thatσ2(A) = λi∞i=l < −β, and hence the growth bound of the semigroup TA2(t)could be arbitrarily prescribed.

The above properties of the operator A admit representation of system(23), (24) in the form

x1 = A1x1 + P1f(x1, x2, t) + P1bu(x1, x2, t), t ≥ 0,x1(0) = x0

1, (41)x2 = A2x2 + P2f(x1, x2, t) + P2bu(x1, x2, t), t ≥ 0,

x2(0) = x02. (42)

It should be pointed out that by virtue of P1b ∈ L(U,Rn), the subspace U2 =ker P1b = u ∈ U : P1bu = 0 has the finite co-dimension l [11]. Hence, thereexists a finite-dimensional subspace U1 = Rm such that U = U1 ⊕ ker P1b,and due to (11) the finite-dimensional subsystem (41) takes the form

x1 = A1x1 + B1[h1(x, t) + u1(x, t)], (43)

where the partition h(x, t) = h1(x, t) + h2(x, t),

u(x, t) = u1(x, t) + u2(x, t) (44)

of the exogenous signals h1(x, t) ∈ U1, h2(x, t) ∈ U2, u1(x, t) ∈ U1, u2(x, t) ∈U2 is used; B1 = P1b|U1 , and the matrix pair A1, B1 turns out to be

212 Y. Orlov

controllable, because the pair A, b, otherwise, would not be exponentiallystabilizable.

The solution to the afore-mentioned rejection problem is based on thedeliberate introduction of sliding modes into the closed-loop system. Fol-lowing the design procedure for controllable finite-dimensional systems pro-posed in [20, Chapter 10] we select such a discontinuity manifold Cx1 = 0,C ∈ Rm×l, det CB1 = 0 that ensures the exponential stability

‖x1(t)‖ ≤ ω‖x1(T )‖e−αt, t ≥ T (45)

of the sliding mode which arises, starting from some time moment T > 0, inthe finite-dimensional system (43) under the control law

u1(x) = −[N(x) + L‖x1‖]Cx1

‖Cx1‖ (46)

where α, ω, L are positive constants and α may be as large as desired. For-mally, in order to specify the matrix C and constant L in an appropriatemanner, one should represent system (43) in the canonical form where thechoice of C and L is straightforward and particularly given in Section 5.The sliding motion in (43) is then governed by the disturbance-independentequation

x1 = [A1 −B1(CB1)−1CA1]x1, (47)

obtained through the equivalent control method by substituting the contin-uous solution

u1eq(x, t) = −(CB1)−1CA1x1 − h1(x, t) (48)

of the equation Cx1 = 0 into (43) for u1. Equation (42) is respectively rewrit-ten as follows:

x2 = A2x2 −B21(CB1)−1CA1x1 ++B2[u2(x, t) + h2(x, t)], t ≥ T (49)

where B21 = P2b|U1 , B2 = P2b|U2 . Due to (40), (45), the unforced system(47), (49) under the zero exogenous inputs u = f = 0 is exponentially stable,and it remains to employ the results of Section 3.1 to reject the externaldisturbance h2(x, t). Setting WA2 =

∫ ∞0

T ∗A2

(t)TA2(t)dt, we design the secondcomponent

u2(x, t) = −N(x)B∗

2WA2x2

‖B∗2WA2x2‖ (50)

of the control u(x, t) in the unit form (40) that imposes the desired robustnessproperty on the closed-loop system.

If the operator A is compact resolvent then, as mentioned earlier, thegrowth bound −β of the semigroup TA2(t) and consequently, the value of


‖W0‖−1 could be specified arbitrarily large in magnitude. Combining thiswith (15), (45) would guarantee the desired decay rate of the closed-loopsystem (41), (42), (46), (50) whenever the pair A, b is approximately con-trollable. Summarizing, the following theorem has been proven.

Theorem 1. Let the unstable part of the unforced dynamics of (10) underu = f = 0 be finite-dimensional and let the pair A,B be exponentiallystabilizable. Then the uncertain infinite-dimensional system (10) is exponen-tially stabilizable by the discontinuous unit controller (44), (46), (50) whichimposes the robustness property with respect to admissible perturbations (11),(12) on the closed loop system. Furthermore, if A has compact resolvent andthe pair A, b is approximately controllable, then the decay rate of the closed-loop system may be specified to be as large as desired.

Example 2: To support the above result by an example let us consider adistributed parameter system described by the parabolic partial differentialequation

∂Q(y, t)/∂t = ∂2Q(y, t)/∂y2 + b(y)[u(Q, t) + h(Q, t)] t > 0,Q(y, 0) = Q0(y), 0 ≤ y ≤ 1 (51)

with Dirichlet boundary conditions

Q(0, t) = Q(1, t) = 0, t ≥ 0. (52)

The boundary-value problem (52) describes the propogation of heat in ahomogeneous one-dimensional rod with fixed temperature at both ends. HereQ(y, t) is the value of the temperature field of the plant at time moment tat point y along the rod, Q0(y) is a scalar twice continuously differentiableinitial distribution which satisfies the boundary conditions (52); u(Q, t) is ascalar control function; h(Q, t) is a scalar unknown disturbance to be rejected,an upper estimate N(Q) ∈ C1 of which is known a priori; b(y) is a scalarquadratically integrable function, all Fourier coefficients of which are nonzero.Clearly,

b(y) =

1d−c if y ∈ [c, d]0 otherwise,

which corresponds to a spatially uniform temperature input over an interval[c, d] ⊂ [0, 1], satisfies the above assumptions if sinπic − sinπid = 0 for alli = 1, 2, . . ..

It is required to design a feedback control law which imposes the desireddecay rate −α as well as robustness with respect to matched disturbances onthe closed-loop system.

If along with the operator b of the multiplication by the function b(y) ∈L2(0, 1) we introduce the operator A = −∂2/∂y2 of double differentiationwith the dense domain

D(A) = ξ(y) ∈ L2(0, 1) : ∂2ξ(y)/∂y2 ∈ L2(0, 1), ξ(0) = ξ(1) = 0,

214 Y. Orlov

then the boundary-value problem (51), (52) can be rewritten as the differ-ential equation (10) in the Hilbert space L2(0, 1). The operator A generatesa strongly continuous semigroup and is compact resolvent [2]. Furthermore,the pair A, b is approximately controllable by virtue of the assumption onthe function b(y). Hence Theorem 1 is applicable to systems (51), (52).

In order to design a unit control-based solution to the stabilization prob-lem stated above, let us select such a number n > 1 that π2(n + 1)2 ≥ αand decouple the spectrum −(πi)2∞i=1 = −(πi)2ni=1 + −(πi)2∞i=n+1 ofA into two parts. Then

H1 = spansin(πiy)ni=1,H2 = sin(πiy)∞i=n+1, U1 = R1, U2 = 0,A1 = diag−(πi)2 ∈ Rn×n, B1 = (P 1b(·), . . . , Pnb(·))T ,

P ib = 2∫ 1

0

b(y) sin(πiy)dy, i = 1, . . . , n,

x1(t) = (P 1Q(·, t), . . . , PnQ(·, t))T ∈ H2.

By virtue of the special choice of the row C = (C1, . . . , Cn) and constant L in(46) the desired decay rate (45) and robustness with respect to the matcheddisturbances are imposed on the closed loop-systems (46), (51), (52) in thefinite-dimensional subspace H1. Since the internal dynamics in H2 is of thedesired decay rate by construction then due to the triviality of the subspaceU2 the resulting control law (44) consists of the first component (46) only.Thus, the unit controller

u(Q) = −[N(Q) + L√Σn

i=1(P iQ)2]sign(Σni=1CiP

iQ)

gives a solution to the stated stabilization problem.Remark 1. To extend the above solution to the case when a point-wise

action b(y)u(Q) = δ(y−y0)u(Q), y0 ∈ (0, 1) is under consideration, the stateequation (10) should be interpreted in another Hilbert space (e.g., a Sobolevspace) where multiplication by the Dirac function is a bounded operator (see[10] for details).

5 Disturbance Rejection in Minimum Phase System

The problems considered in this subsection are to make the output

z(t) = (s, x(t)), s ∈ H (53)

of the uncertain system (10) converge to zero as fast as desired and to ascer-tain conditions which ensure exponential stability of the closed-loop system.For the sake of simplicity, the development is confined to the scalar output,however, the extension to systems with arbitrary finite-dimensional outputis straightforward.


Throughout this section we assume that system (10), (53) has a finiterelative degree

r := mini = 1, 2, ... : b∗A∗i−1s = 0

and s ∈ D(A∗r

). It follows then that

zi(t) := z(i−1)(t) = (A∗i−1s, x(t)), i = 1, 2, . . . , r,

z(r)(t) = (A∗r

s, x(t)) + (A∗r−1s, f(x(t), t) + bu(x(t), t)). (54)

Partitioning the state space as

H = H1 ⊕H2, H1 = spanA∗i−1sri=1,

H2 = x ∈ H : (A∗i−1s, x) = 0, i = 1, 2, . . . , r

and utilizing (54), let us represent the original system (10), subject to thematched disturbance (11), in terms of x1(t) = Σr

i=1zi(t)A∗i−1

s ∈ H1 andx2(t) ∈ H2 as

z1(t) = z2(t), . . . , zr−1(t) = zr(t),

zr(t) = (A∗r

s, x(t)) + (b∗A∗r−1s, u1(x, t) + h1(x(t), t), (55)

x2 = A21x1 + A22x2 + B21(u1(x, t) + h1(x, t)) +B22(u2(x, t) + h2(x, t)) (56)

where

U = U1 ⊕ U2, U1 = spanb∗A∗r−1s, U2 = u ∈ U : (b∗A∗r−1

s, u) = 0,ui, hi ∈ Ui, B2i = P2b|Ui

, A2i = P2A|Hi, ı = 1, 2,

P2 is the projector on H2. It should be noted that the operator A21, definedeverywhere in H1 (A21 is a densely defined operator on the finite-dimensionalspace H1 and hence D(A21) = H1 ), is bounded.

The solution of the above problem presented here is based on the delib-erate introduction of sliding modes in the manifold

cx = Σri=1ci(A

∗i−1s, x) = 0 (57)

where parameters

cr = 1, cr−1 = −Σr−1i=1 µi, cr−2 = Σi<kµiµk, . . . ,

c1 = (−1)r−1Πr−1i=1 µi

are specified to place the roots of the characteristic polynomial of the equation

Σri=1cizi(t) = Σr

i=1ciz(i−1)(t) = 0 (58)

in the open left-half plane at the desired locations µi, i = 1, ..., r − 1. Wedemonstrate that the discontinuous unit control law

u1(x) = −M(x)b∗A∗r−1s

‖b∗A∗r−1s‖sign(cx) (59)

216 Y. Orlov

with

M(x) = γ + N(x) +‖A∗r

s‖‖b∗A∗r−1s‖‖x(t)‖ +

+Σr−1

i=1 |ci||(A∗i

s, x)|‖b∗A∗r−1s‖ , γ > 0 (60)

drives system (10) to the discontinuity manifold (57) within a finite timeinterval. Indeed, differentiating the functional V (t) = 1

2 (cx(t))2 along thetrajectories of (55) and utilizing (12), (55), (59), (60), we obtain

V (t) = cx(t)cx(t) = cx(t)Σr−1i=1 cizi+1(t) +

< A∗r

s, x(t) > + < b∗A∗r−1s, h1(x(t), t) > −

‖b∗A∗r−1s‖M(x)sign(cx(t)) ≤ −2γ

√V (t),

that gives rise to (58) for t ∈ [T,∞) where T = γ−1√V (0). In order to

reproduce this conclusion, one should note that for all t ≥ 0 arbitrary solu-tions V (t) to the latter inequality is majored V (t) ≤ V0(t) by the solution tothe differential equation V0(t) = −2γ

√V0(t), initialized with the same initial

condition V0(0) = V (0). Since V0(0) = 0 for all t ≥ T , V (t) vanishes after thefinite time moment T .

Thus, starting from the time moment T = γ−1√V (0), in the finite-

dimensional system (55), driven by the unit control signal (59), there ap-pears the sliding mode (58), which results in the desired decay rate −α =max1≤i≤rRe µi of the variable x1(t) = Σr

i=1zi(t)A∗i−1

s ∈ H1:

‖x1(t)‖ ≤ ω‖x1(T )‖e−αt, t ≥ T, ω = const. (61)

In order to derive the sliding mode equation (58) one needs to substitute thecontinuous solution

u1eq(x, t) = −Σri=1cizi+1 + (A∗r

s, x)‖b∗A∗r−1s‖ − h1(x, t)

of the equation cx(t) = 0 into (55) for u1. By the same substitution equation(56) is rewritten as follows:

x2 = Ax2 + A21x1 − B21[Σri=1cizi+1 + (A∗r

s, x1)]‖b∗A∗r−1s‖ +

B22[u2(x, t) + h2(x, t)], t ≥ T (62)

where Ax2 = A22x2 − B21(A∗r

s,x2)

‖b∗A∗r−1s‖ .

If the operator A generates an exponentially stable semigroup then dueto (61) and boundedness of A21, the second control component

u2(x, t) = −N(x)B∗

22

∫ ∞0

T ∗A

(t)TA(t)dtx2

‖B∗22

∫ ∞0

T ∗A

(t)TA(t)dtx2‖, (63)


similar to (50), rejects the external disturbance h2(x, t) and ensures the ex-ponential stability of the closed-loop system with the same line of reasoningas in Section 4. Apparently, A generates an exponentially stable semigroupiff the input-output system (10),(53) is exponentially minimum phase.

Thus, the following result has been shown.

Theorem 2. Let s ∈ D(A∗r

) and let system (10), (53) be exponentially min-imum phase and of the finite relative degree r. Then the uncertain system (10)is exponentially stabilizable by the composition u(x) = u1(x) + u2(x) of theunit controllers (59), (63) and the closed-loop system is robust with respectto external disturbances (11), (12).

We conclude this section with two control problems for heat processesand distributed mechanical oscillators which demonstrate the constructiveutilities of the above theorem.

Example 3: Let us modify Example 2 and replace the boundary condi-tions (52) by the appropriate Neumann boundary conditions

∂Q(0, t)/∂y = ∂Q(1, t)/∂y = 0, t ≥ 0. (64)

These conditions appear to describe the propogation of heat in a one-dimensio-nal rod, insulated at both ends. The Fourier coefficients of the functionb(y) are no longer assumed to be nonzero, with the only exception being∫ 1

0b(y)dy. The operator A = −∂2/∂y2 of double differentiation is now de-

fined in D(A) = ξ(y) ∈ L2(0, 1) : ∂2ξ(y)/∂y2 ∈ L2(0, 1), ∂ξ(0)/∂y =∂ξ(1)/∂y = 0 and the boundary-value problem (51), (64) is still representedas the differential equation (10) in the Hilbert space L2(0, 1). Since the spec-trum σ(A) = −(πj)2∞j=0 of A contains zero eigenvalues the unforced system(51), (64) under u = h = 0 is not asymptotically stable.

Specifying the system output (53) as the average temperature

z(t) =∫ 1

0

Q(y, t)dy (65)

of the plant, one can check that the input-output system (51), (64), (65)satisfies all the assumptions of Theorem 4. Indeed, s = 1 and hence s ∈D(Al) for the self-adjoint operator A and arbitrary integer l. Furthermore,differentiating (65) with respect to t along the solutions of (51), employingintegration by parts and applying the boundary conditions (64) yields

z(t) = (u + h)∫ 1

0

b(y)dy

with∫ 1

0b(y)dy = 0, which proves that system (10), (53) is of the relative

degree r = 1. Finally, representing the solution

Q(y, t) =∫ 1

0

G(y, ξ, t)Q0(ξ)dξ +∫ t

0

∫ 1

0

G(y, ξ, t− τ)b(ξ)dξ[u(Q, τ) + h(Q, τ)]dτ

218 Y. Orlov

of the Neumann boundary-value problem (51), (64) via the Green function

G(y, ξ, t) = Σ∞j=0 exp−(πj)2t cosπjy cosπjξ,

one can show the exponential stability of the zero dynamics

Qz(y, t) = Σ∞i=1

∫ 1

0

cos(πiξ)Q0(ξ)dξ exp−(πi)2t cos(πiy)

of (51), (64), (65), written under appropriate initial conditions such that∫ 1

0Q(y, 0)dy = 0, and the suitable control signal u(Q, t) = −h(Q, t) produces

the system output identically zero.Thus, Theorem 2 is applicable to systems (51), (64), (65). According to

the theorem, the controller

u(Q) = − N(Q)∫ 1

0b(y)dy

sign z(t), (66)

imposes a sliding mode along the manifold z = 0 so that the closed-loopsystem is exponentially stable and robust to the matched disturbances.

Remark 2. If the output of the system is replaced by the average temper-ature over an interval [c, d] ⊂ [0, 1] such that

∫ d

cb(y)dy = 0, i.e., the output

is given as

z1(t) =1

d− c

∫ d

c

Q(y, t)dy, (67)

Theorem 2 is still applicable to the input-output system (51), (64), (67).Indeed, the assumption s ∈ D(A∗r

) is now satisfied in the Hilbert spaceL2(c, d) rather than in L2(0, 1) (although such a modification of Theorem 2requires a separate justification, however, the line of reasoning used in theproof of Theorem 2 applies here as well). Moreover, the exponential stabilityof the zero dynamics Qz1(y, t), which is now governed by

∂Qz1(y, t)/∂t = ∂2Qz1(y, t)/∂y2 + ∂Qz1(c, t)/∂y −−∂Qz1(d, t)/∂y, ∂Qz1(0, t)/∂y = ∂Qz1(1, t)/∂y = 0, (68)

is straightforwardly shown through the mode representation

Qz1(y, t) = Σ∞j=0qj(t) cosπjy

where qi(t), i = 1, 2, . . . satisfy qi(t) = −(πi)2qi(t), whereas

q0(t) = −Σ∞i=1

sin(πid) − sin(πic)πi(d− c)

qi(t)

by virtue of z1(t) = 0. Finally, differentiating (67) yields z1(t) = Q′(d, t) −Q′(c, t) + (u + h)

∫ d

cb(y)dy, thereby proving that (51), (64), (67) is of the


relative degree r = 1. Thus, in accordance with Theorem 2, the control law

u(Q) = − N0(Q)∫ d

cb(y)dy

sign z1(t),

N0(Q) > N(Q) + |Q′(d, ·) −Q′(c, ·)|, (69)

which ensures a sliding mode along the manifold z1 = 0, also exponentiallystabilizes the heat process (51), (64).

Remark 3. It is plausible that the output feedback

u(z) = − M∫ 1

0b(y)dy

sign z(t)

with a sufficiently large constant M > 0 still drives the system to the dis-continuity manifold z = 0 and consequently, imposes the desired dynamicproperties as well as robustness with respect to the matched disturbances onthe closed-loop system. Note that the same target can also be achieved byemploying a lumped, e.g., point-wise actuator (see, Remark 1).

Example 4: Let a distributed parameter system be governed by thehyperbolic partial differential equation

∂2Q/∂t2 = ∂2Q/∂y2] − 2∂Q/∂t ++b(y)[u(Q, t) + h(Q, t)], 0 < y < 1, t > 0,

Q(y, 0) = Q0(y), ∂Q(y, 0)/∂t = Q1(y), 0 ≤ y ≤ 1, (70)

subject to the Neumann boundary conditions (64). The boundary-value prob-lem (64), (70) describes the oscillations of a homogeneous string, insulatedat both ends, where the state vector consists of the deflection Q(y, t) of thestring and its velocity Q(y, t) at time moment t ≥ 0 and location y along thestring. The initial distributions Q0(y), Q1(y) are twice continuously differen-tiable functions which satisfy the boundary conditions (64); b, and u, and has well as the operator A and the output z, utilized below, are the same asin Example 3.

If we introduce the operator

A =[

0 IA −2

]

then the boundary-value problem (64), (70) can be represented as the differ-ential equation (10) in the Hilbert space L2(0, 1) ⊕ L2(0, 1). The operator Agenerates a strongly continuous semigroup (see, e.g., [2]), however, the spec-trum σ(A) = −(πj)2∞j=0 of A contains zero eigenvalues and therefore theunforced system (64), (70) under u = h = 0 is not asymptotically stable.

Verification of the assumptions of Theorem 2 is similar to that of Example3, except that the input-output system (64), (65), (70) is of the relative degreer = 2, since z(t) =

∫ 1

0Q(y, t)dy,

z(t) = (u + h)∫ 1

0

b(y)dy

220 Y. Orlov

where∫ 1

0b(y)dy = 0. Thus, by applying Theorem 2 to (64), (65), (70), the

control law

u(Q) = − N1(Q)∫ 1

0b(y)dy

sign z(t) + z(t),

N1(Q) > N(Q) + |z(·)|, (71)

which imposes a sliding mode along the manifold z+ z = 0 (thereby yieldingz(t) → 0 as t → ∞), makes the closed-loop system (65), (64), (70), (71)exponentially stable and robust to the matched disturbances.

Remarks 1-3 remain in force for the stabilization of the distributed oscil-lator (64), (70) as well.

6 Conclusions

Discontinuous control laws are developed for dynamic systems driven in aHilbert space. These control laws impose the desired dynamic properties onthe closed-loop system while retaining the disturbance rejection and robust-ness features similar to those possesed by their counterpart in the finite-dimensional case. The control algorithms proposed do not represent simpleextensions of the finite-dimensional control laws to the infinite-dimensionalcase because the general constructions, quite natural for the finite-dimensionalsystems, become invalid for the infinite-dimensional systems due to the pres-ence of the unbounded operator in the plant equation. The sliding modecontrol theory developed for infinite-dimensional systems is illustrated byapplications to heat processes and distributed oscillators.

References

1. Breger, A.M., Butkovskii, A.G. , Kubyshkin, V.A., and Utkin, V.I. (1980), Slid-ing modes for control of distributed parameter entities subjected to a mobilemulticycle signal. Automation and Remote Contr. 41, 346-355

2. Curtain, R. F. and Pritchard, A. J. (1978) Infinite-dimensional linear systemstheory. Lecture notes in control and information sciences, Springer-Verlag, Berlin

3. Dorling, C.M. and Zinober, A.S.I. (1986) Two Approaches to Sliding Mode De-sign in Multivariable Variable Structure Control Systems. International Journalof Control 44, 65-82

4. Drazenovic, B. (1969) The Invariance Conditions for Variable Structure Systems.Automatica, 5, 287-295

5. Foias, C., Ozbay, H., and Tannenbaum, A. (1996) Robust Control of InfiniteDimensional Systems: Frequency Domain Methods. Springer-Verlag, London

6. Friedman, A. (1969) Partial Differential Equations. Holt, Reinhart, andWinston,New York

7. Gutman, S. (1979) Uncertain dynamic systems - a Lyapunov min-max approach.IEEE Trans. Autom. Contr. AC-24, 437-449


8. Gutman, S. and Leitmann, G. (1976) Stabilizing Feedback Control for DynamicSystems with Bounded Uncertainties. Proceedings of IEEE Conference on De-cision and Control, 94-99

9. Henry D. (1981) Geometric theory of semilinear parabolic equations. Lecturenotes in math. Springer-Verlag, Berlin

10. van Keulen, B. (1993) H∞-Control for Distributed Parameter Systems: A State-Space Approach. Birkhauser, Boston

11. Kirillov, A.A. and Gvishiani, A.D. (1982) Theorems and Problems in FunctionalAnalysis. Springer-Verlag, New York

12. Orlov, Yu.V. (1983) Application of Lyapunov method in distributed systems.Automation and Remote Control44,426-430

13. Orlov, Yu.V. (1993) Sliding mode - model reference adaptive control of dis-tributed parameter systems. Proc. 32nd IEEE Conf. on Decision and Control,2438-2445

14. Orlov, Yu.V. (2000) Sliding mode observer-based synthesis of state-derivativefree model reference adaptive control of distributed parameter systems. Journalof Dynamic Systems, Measurement, and Control, 122, 725-731

15. Orlov, Yu.V. (2000) Discontinuous unit feedback control of uncertain infinite-dimensional systems. IEEE Trans. Autom. Contr. AC-45, 834-843

16. Orlov, Yu.V. and Utkin, V.I. (1982) Use of sliding modes in distributed systemcontrol problems. Automation and Remote Contr. 43, 1127-1135

17. Orlov, Yu.V. and Utkin, V.I. (1987) Sliding mode control in infinite-dimensionalsystems. Automatica 23, 753-757

18. Orlov, Yu.V. and Utkin, V.I. (1998) Unit sliding mode control in infinite-dimensional systems. Applied Mathematics and Computer Science. 8, 7-20

19. Ryan, E.P. and Corless, M. (1984) Ultimate Boundness and Asymptotic Stabil-ity of a Class of Uncertain Systems via Continuous and Discontinuous FeedbackContr. IMA J. Math. Cont. and Inf. 1, 222-242

20. Utkin,V.I. (1992) Sliding Modes in Control Optimization. Springer-Verlag,Berlin

21. Utkin V. I. and Orlov Yu. V. (1990) Sliding mode control of infinite-dimensionalsystems. Nauka, Moscow (in Russian)

22. Zinober, A.S.I., ed. (1990) Deterministic Control of Uncertain Systems. PeterPeregrinus Press, London

23. Zolezzi, T. (1989) Variable structure control of semilinear evolution equations.Partial differential equations and the calculus of variations, Essays in honor ofEnnio De Giorgi, Birkhauser, Boston 2, 997-1018

222 Y. Orlov

Sliding Modes Applications in PowerElectronics and Electrical Drives

Asif Sabanovic1, Karel Jezernik2, and Nadira Sabanovic1

1 Sabanci University, Faculty of Engineering and Natural Sciences Orhanli, 81474Istanbul-Tuzla, Turkey, e-mail: [email protected]

2 University of Maribor, Faculty of Electrical Engineering and Computer ScienceSmetanova ul. 17, SI-2000 Maribor, Slovenia, e-mail: [email protected]

Abstract. Control system design of switching power converters and electrical ma-chines based on the sliding mode approach is presented. The structural similaritiesamong switching converters and electrical machines are used to show that the samestructure of the controller could be used for plants under consideration. The con-troller is designed as a cascade structure with inner current loop designed as asliding mode system with discontinuous control and outer loop (voltage or mechan-ical motion) being designed as a discrete-time sliding mode controller.

1 Introduction

The aim of this paper is to present an application of sliding mode controlin switching power converters and electrical drives. Our intention is to showthat, due to the structural similarities, switching power converters and electri-cal machines could be analysed in the same framework and that the structureof the control system is the same for both plants. The basis for our approachis the analysis of switching converters and electrical machines as the set ofenergy storage elements with their interconnections dynamically changed bythe operation of the switching matrix [2]. The switching matrix plays the roleof a control element determining the power exchange between energy storingelements, introducing change in the structure of the system and, thus mak-ing design in the framework of variable structure systems and sliding modecontrol [1] a natural choice. Engineering methods rather than a historicaloverview of published results will be presented.A functional description of switching power converters and electrical ma-

chines is presented in section 2. As a result of this analysis mathematicaldescription that treats both converters and machines is devised and a for-mulation of the converters and electrical drive control, in the framework ofVSS is derived. In the third section some results of VSS theory that are usedin this paper are reviewed and the control algorithms common in switchingconverters and electrical machines control are discussed. In the same sectiondesign of voltage and power flow control for power converters and design ofthe motion control of electrical machines is presented. In the forth sectionthe design of IM observer is discussed in details. The last section presents


224 A. Sabanovic, K. Jezernik, and K. Sabanovic

experimental example of the neural network realisation of the sliding modesystem for induction machine control.

2 Functional description of switching power convertersand electrical machines

The role of the power converter is to modulate electrical power flow betweenpower sources (Fig. 1). In general power flow can be bi-directional so bothsources may the play role of power generator or power sink (load). The con-verter should enable that interaction of any input source to any output sourceof the power systems. It acts as a link having matrix-like structure. Efficientmodulation of power flow is realised using switch-like elements having zerovoltage drop when conducting and fully blocks current flow when open. Use ofswitches as structural elements of a converter offers opportunity to considera converter as switching matrix. Disregarding the wide variety of designs inmost switching converters, control of power flow is accomplished by varyingthe length of time intervals for which one or more energy storage elements areconnected to or disconnected from the energy sources. Due to the restrictionimposed by Kirchoff’s circuit laws the nature of sources at the input andoutput sides of the switching matrix must be different (voltage or currentsources)[2].For purposes of mathematical modelling the operation of a switch may

be described by a two-valued variable uik(t), (i = 1, ..., n; k = 1, ...,m) ,having value 0 when the switch is open and value 1 when the switch is closedwith average value 0 ≤ uik(t) ≤ 1 . If voltage sources are connected toinput side of the switching matrix then restrictions imposed due to Kirchoff’scircuit laws allow only one switch connecting one of the n input lines to thek−th (k = 1, ...,m) output line can be closed during any time interval, ormathematically

∑n1=1 uik(t) = 1, k = 1, ...,m . Allowed connections will be

referred as permissible combinations.An analogous requirement can be derived if the current source is attached

at the input side of the switching matrix. The operation of the switchingmatrix changes the connections among elements of the switching converter

Fig. 1. Converter as a connection between power sources in a switching matrixconnecting n-dimensional input and m-dimensional output.

Sliding Modes Applications 225

and introduces variation in the dynamical structure of the system. Since therole of switching matrix is to control power flow, the most natural way tomodel the matrix is by introducing a control vector that will represent theeffect of the matrix operation. Let the state of the switches be defined bythe vector sT

sw = [u11 .. uik .. unm] whose elements are two-valued variablesuik(t), (i = 1, ...n; k = 1, ...,m) describing the state of switches in each nodeof the switching matrix [3],[4]. The topological connection of the switchesin the switching matrix is defined by the matrix AM with elements beingfrom the discrete set S3 = 1, 0,−1. The topological connection of the loadwith respect to the switching matrix defines the relations of the variablesat the output lines of the switching matrix to the load quantities and couldbe defined by matrix AL. The operation of the switching matrix can beexpressed by vector u = AMssw. Vector u has a number of distinctive valuesequal to the number of permissible switch connections.

2.1 Common converters and their operational properties

In Table 1 topological structure of the most common converters having volt-age or current sources on input or output sides are depicted with transforma-tion of variables. The simplest matrix - representing DC-to-DC converters -has only two switches interconnecting two unipolar sources. If unipolar andbipolar sources are to be interconnected then the switching matrix must haveat least four switches. If one of the sources to be interconnected is three-phasethen the structure has three output or three input lines, depending on theposition of the AC source. These structures are representing three phase in-verters (DC source at input side) and rectifiers (three phase source at inputside).The role of energy storing elements (L,C) is to balance power flow be-

tween source and sink by temporarily storage and release of energy [2]. Thedynamics of converters depends on the topological relation of the energy stor-age elements to the switching matrix. Further analysis will be concentrate ontwo generic structures - both with inductance energy transfer. In the first- so called buck structure - inductance is connected to the output of theswitching matrix, energy flow from the source is pulsating being modulatedby the switching matrix. In the other - so called boost structure - inductanceis connected to the source and the energy flow from the source is continuouswhile the switching matrix is modulating discontinuous energy flow to theoutput side.

Dynamics of DC-to-DC converters. Buck and boost structures of DC-to-DC power converters are shown in Table 2 along with their mathematicalmodels. In the buck structure the dynamical structure of the system remainsthe same - an LC filter is connected to the variable source. In the case ofthe boost converter the dynamical structure is changed depending on the


Table 1. Topological structure of the most common switching matrices and theirfunctional characteristics.

DC-to-DC converters

Buck structure Common relations Boost structure

vs = vgALu

u = AMssw

AL = [1]

AM = [1 0]

ssw = [u11 u21]T

is = igALu

u = AMssw

DC-to-AC and AC-to-DC single phase converter

Buck structure Common relations Boost structure

vs =1

2vgALu

u = AMssw

AL = [1 − 1]

AM =

[1 0 −1 0

0 1 0 −1

]

ssw = [u11 u12 u21 u22]T is = igALu

u = AMssw

Three phase converters

Inverters Common relations Rectifiers

vs =1

2vgALu

u = AMssw

AL =

1 −1 0

0 1 −1−1 0 1

AM =

1 0 0 −1 0 0

0 1 0 0 −1 0

0 0 1 0 0 −1

ssw =

u11

u12

u13

u21

u22

u23

or

ssw =

u11

u12

u21

u22

u31

u32

vs = vTg ALu

u = AMssw

AL = E

AM =

1 −1 0 0 0 0

0 0 1 −1 0 0

0 0 0 0 1 −1


Table 2. Structures and mathematical models of DC-to-DC converters.

Converter substructure Converter mathematical model

Buck structuredvodt

=iLC

− voRC

diLdt

=Vg

Lu − vo

Lu = AMssw

dvodt

= fv(iL, vo)

diLdt

= fi(vo) + b(Vg)u

Boost structuredvodt

=iLC(1− u)− vo

RCdiLdt

=Vg

L− vo

L(1− u)

u = AMssw

dvodt

= fv(iL, vo) + b1(iL)u

diLdt

= fi(vo, Vg) + b2(vo)u

switch position having either two isolated power storage elements (L andC) or a power filter (L,C) connected to the power source. These facts arereflected in the mathematical description of the converters: buck structuresbeing represented in regular form [5] (system is split in the blocks so thefirst block has the same dimension as control and the second block does notexplicitly depend on control input) (see Fig. 2.a). Boost converter have controlentering both equations as depicted in Fig. 2.b. These features are commonfor converters with depicted position of the switching matrix independent onthe number of input and output lines of the switching matrix.

Fig. 2. Dynamical structure of: a) buck converters and b) boost converters.


Dynamics of single phase DC-to-AC and AC-to-DC converters. Thedynamics of converters with alternative voltage or current sources connectedto either input or output side of the switching matrix is the same as for DC-to-DC converters as illustrated in Table 3. The only difference is in the valuesthat control could take from discrete set S3 = −1, 0, 1 for these convertersand from discrete set S2 = 0, 1 for DC-to-DC converters. This allows toregard a DC-to-AC converter as a structure in which the load is connectedbetween two DC-to-DC converters.

Table 3. Structures and mathematical models of single-phase converters.

Converter substructure Converter mathematical model

dvodt

=iLC

− voRC

diLdt

=Vg

Lu − vo

Lu = AMssw

dvodt

=iLC

u − voRC

diLdt

=Vg

L− vo

Lu

u = AMssw

Dynamics of three phase converters. The switching matrix for all threephase converters DC-to-AC (inverters) and AC-to-DC (rectifiers) is the same.Buck and boost structures for both inverters and rectifiers could easily berecognised for three phase converters as clearly shown in Table 4. In ouranalysis balanced three phase systems is assumed which can be describedin different frames of references: stationary three-phase (a, b, c), orthogonaltwo-phase (α, β) and synchronous frame of references (d, q). Mapping be-tween these frames of references is defined by matrix Aαβ

abc for (a, b, c) to(α, β) and for Adq

αβ to (d, q). In Table 4 mathematical models are presentedin a synchronous frame of references with θr as angular position of the se-lected orthogonal frame of references. Matrix F (θr) = Adq

αβAαβabc is defining

the nonlinear transformation between three phase (a, b, c) and synchronousorthogonal (d, q) frames of references. The (d, q) frame of references is de-termined in such a way that it is synchronous with the three-phase side ofa converter (input side for rectifiers and output side for inverters). In thepresented models notation is used as follows: vT

o = [vod voq] the capacitance


Table 4. Structures and mathematical models of three-phase converters.

Converter substructure Converter dynamics

and control

Adqαβ =

[cos θr sin θr

− sin θr cos θr

]

Aαβabc =

[1 −1/2 −1/20√3/2 −√

3/2

]

[dvoddt

dvoq

dt

]=

[− vod

RC+ ωrvoq

− voq

RC− ωrvod

]+1

C

[1 0

0 1

][iLd

iLq

]

[diLddt

diLq

dt

]=

[− vod

L+ ωriLq

− voq

L− ωriLd

]+

Vg

2L

[1 0

0 1

][ud

uq

]

F (θr) = Adqαβ Aαβ

abc

udq = F (θr)AMssw

dvoddt

= − voRC

+iLC[

diLdt

igq

]=

[− vo

L

0

]+

[Vg

L0

0 iL

][ud

uq

]


abc

udq = F (θr)AMssw


abc

udq = F (θr)AMssw

[dvoddt

dvoq

dt

]=

[− vod

RC+ ωrvoq

− voq

RC− ωrvod

]+

iLC

[1 0

0 1

][ud

uq

]

diLdt

= −vdL

ud − vqL

uq +Vg

L


abc

udq = F (θr)AMssw

dvoddt

= − voRC

+iLdud + iLquq

2C[diLddt

diLq

dt

]=

[ωriLq +

Vg

L

−ωriLd

]− vo2L

[1 0

0 1

][ud

uq

]


voltage vector, iTL = [iLd iLq] inductor current vector and uT = [ud uq]is the control vector, Vg is amplitude of input voltage, R,L,C - converterparameters.The above analysis shows that switching converters can be described as

shown in equations (1)-(3) where for particular converter functions f c(vo, iL),f i(vo, iL), matricesBc(iL, Vg),Bu(iL, Vg) and control u could be determinedfrom the above tables. The DC-to-DC and DC-to-AC single phase convert-ers are SISO systems while three-phase converters are MIMO systems withthe three dimensional control vector. The relation (3) is shown for the pur-pose of having complete definition of the control input in the (d, q) frame ofreferences.

Buck structures Boost structures

dvo

dt= f c(vo, iL)

diLdt

= f i(vo, iL) +Bu(iL, Vg)u

dvo

dt= f c(vo, iL) +Bc(iL, Vg)u

diLdt

= f i(vo, iL) +Bu(iL, Vg)u(1)

usw = AMssw = u (2)


abc , udq = F (θr)AMssw

Adqαβ =

[cos θr sin θr

− sin θr cos θr

], Aαβ

abc =

[1 −1/2 −1/20√3/2 −√

3/2

](3)

Dynamics of electrical machines. To realise necessary power flow aswitching converter (or at least a switching matrix) must be inserted be-tween the source and a machine. Since the electrical subsystem of machinesare modelled as predominantly inductive, the voltage source shall be used atthe source side of the switching matrix. Mathematical models of the mostcommon machines connected to a switching matrix are shown in Table 5.Mechanical motion is the same for all rotating machines and is described asa second order system with electromagnetic torque as the input. The elec-tromagnetic subsystem depends on the magnetic circuitry of the particularmachine and can be described by a simple first order equation for DC ma-chines, by a second order system for PM machines and by a fourth ordersystem for induction machines. Descriptions of the electromagnetic subsys-tem of a machine in Table 5 are presented in (d, q) synchronous orthogonalframe of references with orientation for all of the machines under considera-tion along the rotor flux vector (so called field orientation). The notation is


Table 5. Structure and mathematical models of common electrical machines

Dynamics of electromagnetic system

DC machine supplied from DC source

diqdt

= −R

Liq − Ke

Lω +

Vg

Luq , u = AMssw

DC machine supplied from three-phase source

diqdt

= −R

Liq − Ke

Lω +

Vg

Luq , u = F (θr)AMssw

iq(machine) = id(converter) , iq(converter) = id(machine)(power factor)

PM three-phase machine supplied from DC source

[diddt

diqdt

]=

[−R

Lid − ωiq

−RL

iq − ωid − KeL

ω

]+

[1L0

0 1L

][ud

uq

], u = F (θr)AMssw

Induction machine supplied from DC source

[diddt

diqdt

]=

[−a1id + a3ωrΨq − a2Ψd

−a1iq − a3ωrΨd + a2Ψq

]+

[Kid 0

0 Kidq

][ud

uq

]; u = F (θr)AMssw

[dΨddt

dΨq

dt

]=

[−ωΨq −a4Ψd

−a4Ψd ωΨd

]+

[KΨ 0

0 KΨ

][id

iq

];

Kid = Kiq =1

σmLs

KΨ =RrLr

Lm

a1 =1

σmLs

(Rs +Rr(

Lm

Lr)2)

, a2 =Rr

σmLs(Lm

Lr)2 , a3 =

Lm

σmLsLr,

a4 =Rr

Lr, σm = 1− Lm

LsLr

Dynamics of mechanical subsystem for all machines

[dθdt

dωdt

]=

[ω

−TL(θ,ω,t)J

]+

[0

KT (id)

]iq

as follows: θ, ω angular position and speed of machine, ΨT = [Ψd Ψq] is rotorflux vector; iT = [id iq] is stator current vector; uT = [ud uq] is the controlvector; Vg is amplitude of input voltage; J is moment of inertia; TL(θ, ω, t) isload torque; KT (id) is torque coefficient which depends on the d -componentof stator current; Ke, Kiq, Kid, KΨ are coefficients depending of the machineparameters and flux, Rr ,Rs are rotor and stator resistance, Lr ,Ls are rotor


Fig. 3. Dynamical structure of electrical machines: a) DC machines and b) induc-tion three phase machines.

and stator inductance, Lm mutual inductance, σm leakage factor. For a DCmachine supplied from a three-phase rectifier the selection of the dq frameis related to supply voltage and is reflected in the change of the d and qcoordinates in comparison with dc machine supplied from the DC source. Amodel of a DC machine is given for machines without field winding.The dynamical structure of a machines is presented in Fig. 3. The differ-

ence among DC and AC machines is in the structure of the switching matrixand thus the dimensionality of the control input. For an induction machine,the dynamics of the rotor flux, with currents as input and rotor flux vector asoutput, should be added to the structure. A mathematical model similar tothe one describing buck switching power converters could be used to describethe dynamics of electrical machines [6]

didqdt

= f i(z, idq,Ψ ) +Bu(idq,z,Ψ )u ,

dz

dt= fz(z, idq) ;

dΨ

dt= fφ(Ψ ,z, idq) . (4)

The elements of vector z are the angular position and the angular velocityof the machine. The third equation describing the change of the rotor flux ispresent only for the induction machine. The control input for DC machinessupplied through DC-to-DC or single-phase converters is scalar. If the DCmachine is supplied from a three phase source than the control input is thesame as for three phase rectifiers and three phase inverters and it has theform as given in (5):

udq = F (θr)AMssw ,

F (θr) =[cos θr sin θr-sin θr cos θr

] [1 −1/2 −1/20√3/2 −√

3/2

], (5)

where θr is the position of the synchronous frame of reference.


3 Control of power converters and electrical machines

The goal of the control system design of switching converters, in most of thepractical cases, is reduced to the requirement that the output voltage of theconverter tracks its reference while satisfying certain dynamical constrains(overshoot, load rejection etc.). The same is true for electrical machines forwhich the control goal could be stated as the requirement to have trackingin the torque, or velocity or angular position. In the following sections theunified approach to the control of switching converters and electrical machinesbased on the introduction of sliding mode in the control system will be shown.First we will briefly discuss some results in sliding mode design applicable toswitching converters and electrical machines control.

3.1 Some results in sliding mode control

Variable structure systems are originally defined for dynamic systems de-scribed by ordinary differential equations with a discontinuous right handside. In such a system so-called sliding mode motion can result. This motionis represented by the state trajectories in the sliding mode manifold and highfrequency changes in the control. For sliding mode applications the equa-tions of motion and the existence conditions are two basic questions to bediscussed. Since models of switching converters and electrical machines arelinear with respect to control further analysis will be restricted to the systemsdefined in the following form

x = f(x, t) +B(x, t)u , (6)

where B(x, t) is an n × m matrix, x ∈ Rn, u ∈ Rm. For such a systemboundary-layer regularisation [1],[7] enables the substantiation of the so-called equivalent control method. In accordance with this method, in (6)control should be replaced by the equivalent control, which is the solutionto σ = Gf(x, t) +GB(x, t)ueq = 0, G = ∂σ/∂x, where σ = 0, σ ∈ Rm

is defining sliding mode manifold while σi = 0 describe the so-called switch-ing surfaces. For detGB = 0 equivalent control is ueq = −(GB)−1Gf , thesliding mode equation in the manifold σ = 0 is

x =(E − (GB)−1G

)f , σ = 0 . (7)

From σ = 0, m components x2 ∈ Rm of the state vector x may be foundas a function of the rest (n−m) components x1 ∈ Rn−m as x2 = −σ0(x1),σ0 ∈ Rm and the order of the sliding mode equation (7) may be reduced bym:

x = f1 (x1,−σ0(x1)) , f1 ∈ Rn−m . (8)

For a system subject to disturbances h(x, t) it has been shown [12] that ifh(x, t) ∈ rangeB the sliding mode motion is independent of the disturbanceh(x, t) [6].


To derive the sliding mode existence conditions in analytical form thestability of the projection of the system motion on subspace σ

σ = Gf(x, t) +GB(x, t)u (9)

should be analysed. If GB is an identity matrix system (9) is decomposed onm first order systems and selecting the control such that signs of each compo-nent σi and its derivative are opposite the sliding mode motion will occur ineach discontinuity surface. Other procedures and selection of the Lyapunovfunctions for VSS are discussed in details in [1]. The most interesting factis that the Lyapunov function affirming the convergence to the sliding modemanifold is finite function of time. It vanishes after the finite time interval tes-tifying that sliding mode motion arises in a finite time instant. Sliding modeequations (8) and existence conditions constitute the basis for the variety ofdesign procedures in VSS. To demonstrate some of the design procedures,let us write system (6), subject to disturbance Dh = Bλ, in the so-calledregular form

x1 = f1(x1,x2) ,x2 = f2(x1,x2) +B2u+B2λ , (10)

where x1 ∈ Rn−m, x2 ∈ Rm, f1(x1,x2), and f2(x1,x2) are vectors ofappropriate dimensions and rankB2 = rankB = m. Assume that the slidingmode manifold is defined as σ = σ0(x1) + x2 = 0, σ ∈ Rm. Then theequivalent control is expressed as ueq = −B−1

2 (G1f1(x1,x2) + f1(x1,x2))−λ. It depends on disturbance and in most cases its realization is unpractical.Calculating (u+ λ) from dσ/dt = 0 and substituting it to second equationsin (10) then, when the sliding mode appears in this manifold, the systembehaviour is governed by the (n−m) order equation

x1 = f1 (x1,−σ0(x1)) , x2 = −σ0(x1) . (11)

In (11) vector σ0(x1) could be treated as ”virtual control” and should beselected to satisfy the desired system dynamics. For control input selectionthe projection of the system motion on m-dimensional space σ is found

σ = f(x1,x2,λ) +B2u, f ∈ Rm ,

f(x1,x2,λ) = G1f1 + f2x2 +B2λ . (12)

The discontinuous control u = −B−12 Msign(σ), M = const. > 0 leads to

σi = fi(x1,x2,λ)−Mui, i = 1, ...,m . (13)

There exists large enough M > 0 such that the functions σi, i = 1, ...,mand derivatives σi have opposite signs, sliding mode will occur in each of thediscontinuity surfaces.Discrete time sliding mode was introduced for discrete time plants [8],[9],[10].

The most significant difference with the continuous time sliding mode is that


motion in the sliding mode manifold may occur in discrete time systems withcontinuous right hand side By applying the sample and hold process withsampling period T , and integrating the solution over interval t ∈ [kT, (k+1)T ]with u(t) = u(kT ) and d(t) = d(kT ), the discrete time model of plant (6)may be represented as

xk+1 = F kxk + Buk + Pdk . (14)

The sliding manifold is defined as σk = Gxk, k = 1, 2, . In [8] the equivalentcontrol ueq

k = ueqk (kT ) is defined as the solution of

σk+1 = Gxk+1 = GF kxk +GBueqk +GPdk = 0 . (15)

Provided that detGB = 0 the equivalent control can be expressed asueq

k = −(GB)−1G(F kxk + Pdk) . (16)

It is important to note that matching conditions now are defined in terms ofmatrices G, B, P . The required magnitude of control (16) may be large andlimitation should be applied, so the final form of the control is

ueqk =

−(GB)−1G(F kxk + Pd∗k), if |uk| < Uo

−Uosign(σk), if |uk| ≥ Uo, (17)

where d∗k is the estimated disturbance and Uo is a control input bound. In an-

other approach for system (6) asymptotic stability of the solution σ(x) = 0can be assured if one can find a control input such that the stability crite-ria are satisfied for the following Lyapunov function ν = σTσ/2 with therequirement that the time derivative (dν/dt) has a certain form, for exam-ple dν/dt = −σTDσ, D > 0, [11]. Then the control input, with samplinginterval T , that satisfy the given requirements is in the form

uk = uk−1 − (GBT )−1 ((E + TD)σk − σk−1) . (18)

The realisation of control (18) requires information on the sliding functionsand the plant gain matrix, which is much easier to obtain than informationnecessary to implement algorithm (17).Mathematical models of the switching converters and electrical machines

could be presented in regular form (10) with discontinuous control influencingthe change of the currents and currents being treated as ”virtual control” inthe voltage dynamics (for converters) or mechanical motion dynamics (formachines). The structure of the boost converters is more complicated withcontrol entering all the equations of the system. Despite the differences in thedynamical structure the control system design for buck and boost convertersand electrical machines may follow the two-step procedure:

• Select control u such that inductor current (or electromagnetic torque inelectrical machines) tracks its reference;


• Select the current reference (virtual control) so that capacitance voltage(or mechanical coordinates) satisfy prescribed dynamical behaviour.

This procedure is not so obvious for the boost structures since control en-ters both equations. In the framework of sliding mode systems the aboveprocedure for boost converters requires substitution of the equivalent controlto the first equation and then taking current reference as ”virtual control”input. In the following sections we will show the consequent application ofthe above procedure in details without presenting unnecessary details forparticular converters or machines.

3.2 Control of DC-to-DC converters and DC machines

In this section the control of DC-to-DC converters and DC machines bothhaving scalar control input is discussed. Control of DC machines suppliedfrom three-phase sources will be discussed in the section dealing with controlof three-phase converters and AC machines.

Control of DC-to-DC buck converter. Assume the current reference ascontinuous function irefL (t) then, for the tracking error σ = irefL (t) − iL(t)and the control selected as u = (1 − signσ)/2 the sliding mode exists if theequivalent control 0 ≤ ueq ≤ 1 is calculated as:

dσ

dt=d(irefL − iL)

dt=direfL

dt+voL

− VgLueq = 0

⇒ ueq = 1Vg(LdirefL

dt+ vo) . (19)

Substituting the equivalent control to the original equations of the systemone can obtain:

dvodt

=irefL

C− voRC

; iL(t) = irefL (t) . (20)

From (20) reference current could be selected using well-established designprocedures for linear systems. For example, if the first order response σv =ρ(vrefo − vo) + d(vrefo − vo)/dt = 0 of closed loop system is required, thereference current could be easily determined as

irefL = sat(voR+ ρC(vrefo − vo) + C dv

refo

dt

)⇒ irefL = sat(iL + Cσv)(21)

where sat(•) is the saturation function. Structure of system (21) is shown inFig. 4. Another structure of the control system may be determined from therequired closed loop dynamics by inserting dvo/dt = iL/C − vo/RC into theexpression for σv which leads to σv = ρ(vrefo −vo)+dvrefo /dt+vo/RC−iL/C =


Fig. 4. Structure of the converter control system.

(irefL − iL)/C and selecting the control as u = (1− sign(Cσv))/2 the slidingmode motion is established in manifold σv = 0 if 0 ≤ ueq ≤ 1 and the sameresult is obtained as in algorithm (21).The structure in Fig. 4 is suitable for the implementation of different

control algorithms in designing the reference current, thus it leaves moreroom for merging other control techniques with the sliding mode. This willbe especially clear in the control of the boost type of power converters.

Control of DC machines. By following the same procedure as for buck con-verters the DC machine tracking error could be defined as σ = irefq (t)− iq(t)with the control u = (1 − signσ)/2 the sliding mode exists if the equivalentcontrol satisfies −1 ≤ ueq = (L/Vg)(direfq /dt+R/Liq+Ke/Lω) ≤ 1. By sub-stituting ueq to the original system the motion of the Dc machine is reducedto the second order system

dθ

dt= ω ;

dω

dt= −TL(θ, ω, t)

J+KT (id)J

irefq . (22)

By requiring the closed loop transient to satisfy σθ = Cθ(θref−θ)+Cωd(θref−θ)

dt

+d2(θref−θ)dt2 = 0 determined by the design parameters Cθ and Cω the refer-

ence current becomes

σθ = Cθ(θref − θ) + Cωd(θref − θ)

dt+d2(θref − θ)

dt2=KT

J(irefq − iq) ,

irefq = sat(TLK(id)

+ Cθ(θref − θ) + Cωd(θref − θ)

dt+d2θref

dt2

). (23)

Note that for Cθ = 0 the above dynamics reduces to σω = Cω(ωref − ω) +Cω(d(ωref −ω)/dt) = 0 and defines the desired dynamics for velocity control.Implementation of control (23) requires information on the machine loadwhich could be obtained using disturbance observer techniques proposed byOhnishi in [13]. Manipulating (23) one can determine a much more simple,way of calculating reference current

irefq = sat(iq +

J

KT (id)σθ

). (24)


Structure of the control system is the same as the one depicted in Fig. 4.These results depict earlier shown similarities between buck converters andelectrical machines. The structure of the control is the same, with necessarychanges in measured variables. In [22], [23] the application of the slidingmode control to DC machine combining the acceleration, the velocity and theposition control is proposed. Proposed algorithm ensures robustness againstparameters and disturbance changes even in the acceleration stage. Designof the sliding mode control for DC electrical machines based on the reducedorder model (dynamics of the electrical current is neglected) is discussed indetails in [4] and [26].

Control of boost DC-to-DC converter. Assume the current referenceas the continuous function irefL (t) then, selecting tracking error σ = irefL (t)−iL(t) and control u = (1 − signσ)/2 the sliding mode exists if equivalentcontrol (25) satisfies conditions 0 ≤ ueq ≤ 1dσ

dt=d(irefL − iL)

dt=direfL

dt+voL(1− ueq)− Vg

L= 0

⇒ ueq = 1vo

(LdirefL

dt+ vo − Vg

). (25)

By substituting ueq to the original equations of the boost converter one coulddetermine

C

2d(vo)2

dt+v2oR= Vgi

refL − L

2d(irefL )2

dt= (Vg − vL)irefL , iL = i

refL , (26)

where vL represents the voltage drop on the inductance L. System (26)may be interpreted as the description of power conservation in the circuit(C/2)d(vo)2/dt + vLi

refL = Vgi

refL − (v2o/R) . From the control point of

view it could be regarded as a linear first order system with the squareof the output voltage v2o as output and reference inductor current as con-trol. With such definition of variables the mathematical description reducesto the first order system nonlinear with respect to the control. For DC-to-DC converters the change of energy stored in inductance could be neglected(average vL = 0) and the system reduces to the first order linear system(C/2)d(vo)2/dt+(v2o/R) = Vgi

refL and selection of the reference current may

follow the same procedure as for buck converters [3],[12]. Another often ap-plied solution is the feed-forward calculation of the reference current fromthe reference voltage [26].

3.3 Control of three phase converters and three phase machines

Three phase converters and electrical machines structurally differ from theirDC counterparts in the number of energy storage elements and in the struc-ture of the switching matrix. The dynamical structure of the systems remains


the same as for their DC counterparts except that the there-phase are MIMOsystems. That allows conduct design of the control in the same two-step pro-cedure as applied for dc systems.

Current control in three phase systems. For modelling and controldesign purposes the three phase switching matrix has been defined by thethree-dimensional control vector u = AMssw . Vector ssw has six elementsbut due to electric circuits constraints for buck inverter and boost rectifier,vector ssw may have the following values S1 = [1 0 0 0 1 1], S2 = [1 1 0 0 0 1],S3 = [0 1 0 1 0 1], S4 = [0 1 1 1 0 0], S5 = [0 0 1 1 1 0], S6 = [1 0 1 0 1 0],S7 = [1 1 1 0 0 0], S8 = [0 0 0 1 1 1]; for boost inverter and buck rectifiervector ssw may have the following values S1 = [1 0 0 0 0 1], S2 = [0 0 1 0 0 1],S3 = [0 1 1 0 0 0], S4 = [0 1 0 0 1 0], S5 = [0 0 0 1 1 0], S6 = [1 0 0 1 0 0],S7 = [1 1 0 0 0 0], S8 = [0 0 1 1 0 0], S9 = [0 0 0 0 1 1].For inverters and rectifiers the number of independent control inputs is

three. As depicted in Fig. 5 for three-phase inverters the number of inde-pendent variables to be controlled is two: for inverters these are the d andq components of output voltage, for AC machines they are the same as thecomponents of supply voltage. For inverters and machines supplied from DCsources there is no variable to be controlled on the input side. For three-phaserectifiers the output is the DC source and thus only one independent variable(output current or voltage) is to be controlled. On the input side of the rec-tifier, the magnitude of voltage or current are defined thus only phase shiftbetween the voltage and the current vector could be controlled. This allowsintroduction of an additional requirement to the control system design. The

Fig. 5. The assignment of the degrees of freedom in control for three-phase switch-ing matrix.


most natural choice is to relate this additional requirement to the selectionof the switching pattern.Let us now look at design of the switching pattern for the three phase

converters in more details. Current control is based on the sliding mode ex-istence in the manifold σT = [iref (t) − i]T = 0 where vector σT = [σd σq]T

with σd = irefd (t) − id, σq = irefq (t) − iq and irefd , irefq are continuous func-

tions to be determined later. Design of the current controller is based on thesystem description (4) didq/dt = f idq +Budqudq where matrix Budq is diag-onal. The structure of function f idq and matrix Budq could be easily foundfrom mathematical models given in Tables 4 and 5. The time derivative ofσT = [σd σq]T is determined as

dσ

dt=direfdq

dt− didqdt

=direfdq

dtf idq − Budqudq, uT

dq = [ud uq] . (27)

Equivalent control can be calculated as B−1udq[di

refdq /dt − f idq] = ueq and

equation (27) is expressed as

dσdq

dt= Budq[ueq − udq(Si)], i = 1, ..., 9 . (28)

Control vectors could take values from the discrete set S = S1,S2,S3,S4,S5,S6,S7,S8,S9 as depicted in Fig. 6.a. All realizable values of the equiv-alent control lie inside the hexagon spaned by the elements of the set S8[6].The rate of change of error is proportional to the differences between thevector of equivalent control and the realisable control vectors. For a partic-ular combination of errors all permissible vectors Si that satisfy the slidingmode existence conditions could be determined from (dσd/dt)σd < 0 and(dσq/dt)σq < 0 or sign(ueq − udq(Si)) = −sign(σdq) as shown in Fig. 6.b.For some combinations of errors there are more than one permissible vector

Fig. 6. Control vectors and selection of permissible control for given combinationof the signs of control errors.


that leads to an ambiguous selection of the control and consequently existenceof more than one solution for the selection of switching pattern .The same could be concluded from rankF (θr) = rank(Adq

αβAαβabc) = 2.

Ambiguity in selection of the control vector based on selected ud and uqallows us to have a number of different PWM algorithms based on satisfyingsliding mode conditions in (d, q) frame of references. In early works [6] thefollowing solution was proposed: add an additional requirement ϑ(t) = 0 tothe control system specification so that (27) is augmented to have the form[ dσdq

dtdϑdt

]=[

diref

dt − f idq

fϑ

]−[

BudqF (θr)bTϑ

]u(Si) , (29)

dσN

dt= fN − BNu(Si), uT (Si) = [ua ub uc] . (30)

Vector bϑ should be selected so that rankBN = 3. the simplest solution is forϑ(t) = ua+ub+uc [6],[26] then matrix BN will have full rank. To determinethe switching pattern, the simplest way is to use the nonlinear transformationσs = B−1

N σN , then the sliding mode conditions are satisfied if the control isselected as

sign (uj(Si)) = −sign(σsj), −1 ≤ ueq ≤ 1 . (31)

This line of reasoning with some variations has been the most popular indesigning the sliding mode based switching pattern [4],[6].Another solution implicitly applied in most of the so-called space vector

PWM algorithms is based on the simple idea [5] using transformation

uabc = rank(AdqαβAαβ

abc)Tudq

to the (a, b, c) reference frame. Then components ua, ub and uc of u = AMssw

are selected according to the following rule

Si =

sign(ua(Si)) = sign(udcosθr − uqsinθr)sign(ub(Si)) = sign(udcos(θr − 2π/3)− uqsin(θr − 2π/3))sign(ub(Si)) = sign(udcos(θr − 4π/3)− uqsin(θr − 4π/3))

i = (1, 2, ..., 8, 9) . (32)

This idea is analysed in details in [6]. Further simplification of this algorithmleads to its implementation using a look-up table [14].In [24] and [25] the so-called space vector PWM based on the sliding mode approach is discussed.The solution is based on the expression (32) but it is realized using spacesectors.In the application of above algorithms switching is realized using hystere-

sis which, as follows from (28) directly determine the current ripple to beequal to the half of the hysteresis width. For the given current ripple (con-stant hysteresis width) the time between two switching for each componentis directly proportional to [ujeq −uj(Si)], (j = d, q), (i = 1, ...9). A new class


of the switching algorithms based on the simple requirement that controlshould be selected to give the minimum rate of change of control error couldbe designed for which the same error will be achieved with less switchingeffort. The algorithm can be formulated in the following form [3]

Si =

min ‖ ueq(t)− u(Si) ‖&

sign [ueqd − ud(Si)] • σd(t) = −1&

sign [ueqq − uq(Si)] • σq(t) = −1

, i = (1, 2, ..., 8, 9) . (33)

The difference in behaviour for algorithm (32) and (33) is depicted in Fig. 7,where the steady state operation of buck inverter current control is shown.The operating point for both algorithms is the same and the width of hys-teresis is also kept the same. The difference in the switching frequency of thevoltage is easily detectable.All of the above algorithms naturally include so-called over-modulation

functionality. This can be seen in the diagrams depicted in Fig. 8. For al-gorithm (33) behaviour of an induction machine (P=4kVA, p=2, U=220 V)current control loop is depicted. (α, β) currents and control are shown inFig. 8.a. for a loaded machine and wref = 50π [rad/s]. Note that switchingis regular between the two closest vectors and zero vector. In Fig. 8.b. thecurrent vector is depicted for wref = 95π [rad/s]. For this condition the slid-ing mode existence conditions are violated in certain regions of the plane.In Fig. 8.c. the current vector is depicted for wref = 100π [rad/s]. For thiscondition the sliding mode existence conditions cannot be satisfied and thesystem operates under a six-step mode.Due to the specifics of three-phase balanced systems the number of inde-

pendent controls for the switching matrix is higher than the dimension of thecontrolled current vector This is the basic reason that three-phase PWM, un-der many different names, is still attractive as a research topic. The solution

Fig. 7. The steady state operationof buck inverter current control forswitching algorithms (32) - lowertrace and (33) - upper trace.

Fig. 8. Change of the current and controlvectors in (α, β) frame of references.


proposed in [15] shows that the formalization of the minimization of TDF inthe above framework has lead to a solution that is far better in comparisonwith other so-called space vector approaches [16].In the sliding mode dynamics of current the control loop is reduced to

σT = [iref (t)−i]T = 0 or irefd (t) = id and irefq (t) = iq with equivalent controlbeing determined as B−1

udq [diref/dt − f idq] = ueq. In order to complete the

design of converters and electrical machines the reference currents shall bedetermined.

Voltage and mechanical motion control system design. For buck con-verters and electrical machines with reference currents interpreted as virtualcontrol inputs the description could be easily transformed to the followingform

dx

dt= fx(x) +Bi(x)iref , (34)

where vector xT = [x1 x2] represent either the vector of output voltage (forconverters) or the vector with velocity and position (for electrical machines).For the buck inverter both components of the reference current could be

determined from the specification of the voltage loop, but for buck rectifiersonly the d-component of the source current can be determined from thevoltage loop specification. The q-component of the source current does notinfluence the output voltage and thus represent current circulating betweensupply sources and creating reactive power flow from sources. The same isdirectly applicable for a DC machine supplied by the three-phase rectifier.Since all machines have the same structure of mechanical subsystem the

results obtained for DC machines may be directly applicable to AC machinesthus giving a way of determining one component of the current vector. Theother component of the current vector should be determined from the require-ment of the magnetic circuits of the machine and is specific for each type ofthe machine. For PM and induction three-phase machines the d-componentof the current defines the rotor flux so it should be selected taking rotor fluxbehavior into consideration.The requirements for converters and machines are presented in Table 6.

The selection of the components of the switching function vector is givenalong with the expression for the reference current calculation. The referencecurrent is selected following the discrete time sliding mode control designand for all systems under consideration it is irefk = irefk−1 − (GBT )−1((E +TD)σk −σk−1); G = ∂σ/∂x where T is the sampling interval. The reali-sation of this control algorithm requires information on the sliding functionsand the plant gain matrix.Consider the three-phase boost rectifier connected as a controllable cur-

rent source as shown in Fig. 9. Its role is to generate current flow so thatthe energy source is loaded by active power or the reactive power of source


Table 6. The selection of desired dynamics and control vector

Type Function σ Reference currents

Buckrectifier

σd = vrefo − vo

σq = irefqav − iqav usually irefqav = 0

Dc machineσd = Cθ∆θ +

d∆θ

dt; ∆θ = θref − θ

σq = irefqav − iqav usually irefqav = 0

Buckinverter

σd = vrefod − vod

σq = vrefoq − voq

i refk = irefk−1 − (GBT )−1

( (E + TD)σk − σk−1);

G = ∂σ/∂x

PMsynchronousmachine

σd = irefdav − idav usually irefdav = 0

σq = Cθ∆θ +d∆θ


irefdq = [irefd irefq ]T

σrefdq = [σd σq]

T

Inductionmachine

σd = σφ(φ, φref ) = ∆φ;

∆φ = φref − φ

σq = Cθ∆θ +d∆θ


Boostrectifier

σd = µ((vrefo )2 − (vo)2

)σq = irefqav − iqav usually irefqav = 0

is zero Qs = 0 which gives irefsq = 0. The d-component reference currentshall be determined from the capacitance voltage. It is easy to show that(C/2) (dv2od/dt) + v

2od/R = Vgi

refd − vLirefd which is the same as for the DC-

to-DC boost converter and all comments regarding the DC-to-DC boost con-verter control are applicable.The structure of the power converters and electrical machines system can

be presented as in Fig. 10. where the current control loop operates in slidingmode with discontinuous control. The structure is the same as one shownin Fig. 4. with additional details on the outer loop controller. The selectedstructure is only one of the several possible solutions, other structures maybe derived by applying some other design procedures many of which aredeveloped in the framework of motion control systems. By doing so, essentialfeatures of the sliding mode are preserved by current loop design.


Fig. 9. Power flow in system when boost converter is used as an active power filter.

Fig. 10. Structure of the converters and electrical machine control system.

Application of the above algorithms requires information on currents andvoltages for converters and mechanical coordinates for electrical machines.Usually measurement of electrical quantities is not considered demandingso realization of the control algorithms in the case of switching convertersdoes not represent any problem. This may not be true for AC electricalmachines and especially for the induction machine. For these machines thesynchronous frame of references is determined by the rotor flux vector, whichis not accessible for measurement and should be derived using observer. Sinceinduction machine is a nonlinear system the observer design may not be sostraight forward a task. In the section 4 of this paper we will be discussing thesliding mode based observers of induction machine rotor flux and velocity.

4 Induction machine observer

Design of a IM sensorless drives is still a challenge. The basic problem is speedestimation especially at the low speed range and under light load conditions.In this section the VSS approach to rotor flux and speed estimation of aninduction machine will be discussed.The description of the machine in the (α, β) frame of references is

dΨ r

dt= −BrΨ r +

RrLm

Lris; Br =

[Rr

Lr−ω

ω Rr

Lr

], (35)

disdt=

1σLs

(Lm

Lr

(−BrΨ r +

RrLm

Lris

)−Rsis + us

). (36)


Formally it is possible to design a stator current observer based on voltageand current measurements and with a rotor flux vector derivative as thecontrol input:

disdt=

1σLs

(−Lm

LruΨ −Rsis + us

). (37)

This selection of the observer control input is different from the usually usedcurrent error feedback (well known Gopinath’s method), application of slid-ing mode control based on current feedback [17], or selecting the unknownvelocity as the control input [18],[4],[26].The estimation error is determined as

dεi

dt=d(is − is)dt

=1σLs

(Lm

Lr((−BrΨ r +

RrLm

Lris) + uΨ )−Rsεi

)(38)

If the sliding mode exists then from dεi/dt = 0, and εi = 0. Under theassumption that the angular velocity is known, from (38) one can find

Ψ r = B−1r

(uΨeq − RrLm

Lris

). (39)

In the observer design suitable for the sensorless drive, the observer controlinput should be a known function of the motor speed so that, after establish-ing sliding mode in current tracking loop, the speed can be determined as aunique solution. This leads to the following selection of the structure of thestator current observer

disdt=

1σLs

(Lm

Lr

(−uφ +

RrLm

Lris

)−Rsis + us

)(40)

and the estimation error becomes:

dεi

dt=d(is − is)dt

=1σLs

(Lm

Lr(−BrΨ r + uφ)−

(RrL

2m

L2r

−Rs

)εi

)(41)

Algorithm (18) could be used to calculate the control uφk = uφk−1 ++(σLsLr/LmT )((1 + ρT )εik − εik−1) with εi = [εiα εiβ ]. From dεi/dt = 0the equivalent control is determined as uφeq = BrΨ r. The rotor flux observercould be selected as having the same structure as (35) with the additionalconvergence term f whose structure will be explained later

dΨ r

dt= −uφ +

RrLm

Lris + f . (42)

Now flux estimation error can be calculated as:

dεΨ

dt=d(Ψ r − Ψ r)

dt= −BrΨ r + uφ +

RrLm

Lris − f . (43)


To ensure convergence f could be selected in the following form:

f = kεΨ , (44)

where[εΨα

εΨβ

]= −

[Ψα Ψβ−Ψβ Ψα

] [µη

], µ =

∆xr xr +∆ω ωω2 + x2r

, η =∆xr ω −∆ω xrω2 + x2r

,

and[ωxr

]=

1‖ Ψr ‖2

[Ψβ −ΨαΨα Ψβ

] [uφαuφβ

], (45)

[∆ω∆xr

]=σmLsLr

LmT

[Ψα −ΨβΨβ Ψα

](ρεi +

dεidt

). (46)

If the sliding mode in the current control loop (41) exists and both ∆ω =0 and ∆xr = 0, then µ = 0 and η = 0 and consequently εΨ = 0. Theconvergence of the observer is easier to analyse by projecting errors in the(d, q) frame of references as given by (47)[edeq

]= −

[Ψα Ψβ−Ψβ Ψα

] [εΨα

εΨβ

]. (47)

After some algebra one can find[ ded

dtdeq

dt

]= −

[ −k Te + ωTe − ω −k

] [edeq

]+ k ‖ Ψ ‖

[εiαεiβ

]. (48)

The design parameter k could be selected from (48) so that the estimatedrotor flux tends to its real value. Te denotes the electromagnetic torque ofthe machine.

5 Neural network application in sliding mode systems

As an example of the application of the above ideas in this section a neu-ral network controller for the induction machine will be discussed [21]. Theprocedure below is valid for all converters and electrical machines since all ofthem could be presented in the form (34). The structure of the system witha neural network controller is depicted in Fig. 11. For system (34) and thesliding mode manifold selected as given in Table 6. The equivalent controlcould be determined as irefeq = −(GB)−1(Gfx). In the system of Fig. 11 theneural network is used to determine the unknown part of equivalent controlin the system. The control input could be expressed as

iref = −(GB)−1(GN(x, t))− (GB)−1Dσ , (49)


Fig. 11. The structure of the control system.

where GN(x, t) is the output of the neural network. This control input givesthe time derivative of the Lyapunov function candidate V = σTσ/2 as [20]

dV

dt= σT dσ

dt= σT (G(fx − N(x, t)))− σTDσ . (50)

The stability conditions will be satisfied if ‖ σTG(fx − N(x, t)) ‖<‖σTDσ ‖ holds, in other words if neural network is trained to make zerothe error function Z = G(f(x, t) − N(x, t)) = Dσ + σ. This means thatthe neural network approximation error could be calculated from the slidingmode function. An obvious selection of the role of the neural network is tominimise

Γ =12ZTZ =

12(Dσ + σ)T (Dσ + σ) (51)

thus to assure mapping GN → Gf . An important feature of the developedscheme is that the approximation error is available on-line. It is important tonotice that the feedback controller assures the stability of the overall system.Assume neural network model

yli = gli

k∑

j=1

wlija

lj + b

li

, gli(netli) = tanh(netli) = 2

1 + e−2netli

− 1 ,

where alj is the j-th input of the l-th layer, yli is the output of the i-th neuron

in the l-th layer, wlij is the weight connecting the j-th input and the i-th

neuron in the l-th layer and bli is the bias of the i-th neuron in the l-th layerof the neural network. Learning proceeds using a gradient descent where ηdenotes the learning rate ∆wl

ij = −η ∂E/∂wlij . For the output layer, denoted

by L, the following equations can be used:

∂Γ

∂yL−li

= G(fx − N(x))(−gm) , (52)

∆wLij = ηgm(Dσ + σ)gl(aLi )y

Lj . (53)


Fig. 12. Experimental system. Fig. 13. The structure of the neural network.

Fig. 14. Desired values of position, ve-locity and acceleration.

Fig. 15. Experimental results for per-forming task presented in Fig. 14.

For the hidden layer a back-propagation learning algorithm as is used with∆wl

ij = η δlj y

lj .

The mechanism featured in Fig. 12 was used in the experiment [21]. Ahammer, mounted on the axis of the induction motor was used and a mass -spring - damper load was added to show effects of coupling. The two layeredneural network used as an on-line estimator is featured in Fig. 13. ϕd is thedesired position and ωd is the desired velocity of the system. The systemtask was to learn to perform the operation, described in Fig. 14. Movementis performed using the so-called sin2 shaped acceleration. The results aredepicted in Fig. 15, and good tracking performance is obtained.

6 Conclusions

A unified approach to control system design for switching converters andelectrical machines is discussed. It has been shown that, due to the struc-tural similarities, switching power converters and electrical machines couldbe analysed in the same framework and the control system structure is thesame for both classes of the plant. The switching matrix plays the role of acontrol element, introducing change in the structure of the system and, thus


making design in the framework of variable structure systems and slidingmode control a natural choice. Engineering methods rather than historicaloverview of published results is presented. As an example of the possibility ofcombining neural networks and VSS approaches the induction machine neu-ral network controller is discussed. Due to the appropriate selection of thesystem’s error the learning procedure has been determined so that it does notrequire any measurements except those needed for the sliding mode functioncalculation. This error signal has been used for the gradient descent algo-rithm. Theoretical results have been confirmed by experiments on IM loadedwith a nonlinear load.

References

1. Utkin, V. I.,(1992) Sliding modes in control and optimization, Springer-Verlag.2. Wood, P., (1981) Switching Power Converters, Van Nostrand Reinholh.3. Sabanovic, N., Ohnishi, K. and Sabanovic, A., (1992) ”Sliding Modes Controlof Three Phase Switching Converters,” Proc of IECON’92 Conference, 319-325,San Diego, USA.

4. Utkin, V. I., (1993) ”Sliding Mode Control Design Principles and Applicationsto Electric Drives”, IEEE Tran. Ind. Electr. Vol. 40, No.1, 421-434.

5. Luk’yanov, A.G. and Utkin, V. I., (1991) ”Methods of reducing equations ofdynamic systems to a regular form,” Aut. Remote Control, Vol. 42, No.12,413-420.

6. Sabanovic, A., Izosimov, D.B., (1981) ”Application of sliding mode to inductionmotor control,” IEEE Trans. Ind. Aut. Vol. IA 17 No.1, 41-49.

7. Drazenovic B., (1969) ”The invariance conditions in variable structure sys-tems”, Automatica, Vol.5, 287-295, Pergamon Press.

8. Drakunov, S. V. and Utkin, V. I., (1989) ”On discrete-time sliding modes”,Proc. of Nonlinear control system design Conf., 273-278, Capri, Italy.

9. Furuta K., (1990) ”Sliding mode control of a discrete system”, System andControl letters, Vol. 14, No. 2, 145-152.

10. Utkin, V.I., (1993) ”Sliding Mode Control in Discrete Time and Difference Sys-tems”, Variable Structure and Lyapunov Control, Ed. by Zinober A.S., SpringerVerlag, London.

11. Sabanovic, A., (1992) ”Sliding modes in motion control systems”, Proc. of 2ndIEEE Intl. Workshop on advanced motion control, 150-156, Nagoya, Japan.

12. Venkataramana, R., Sabanovic, A. and Cuk, S., (1985) ” Sliding Mode Controlof DC-to-DC Converters,” Proc. of IECON’85, San Francisco, USA.

13. Ohnishi , K. and Murakami, T., (1989) ”Application of Advanced Control Tech-niques in Electrical Drives”, Proc. of IEEE Conf. on Microcomputer Control ofElectric Drives, Trieste, Italy.

14. Sabanovic, N., Sabanovic, A. and Ninomiya, T., (1994) ”PWM in Three-PhaseSwitching Converters - Sliding Mode Solution”, Power Electronics SpecialistConference PESC’94, Taipei, Taiwan.

15. Chen, Y., Fujikawa, K., Kobayashi, H., Ohnishi, K. and Sabanovic, A., (1997)”Direct Instantaneous Distorsion Minimization Control for Three Phase Con-verter,” Tran. IEEJ, Vol. 117-D, No. 7, (in Japanese).


16. Holtz, J., Lotzkat, W., Khambadkone, A., (1992) ”On Continuous Control ofPWM Inverters in the Over modulation Range including the Six-Step Mode,”Proc. of IECON’92 Conference, 307-312, USA.

17. Sangwongwanich, S., Doki, S., Yonemoto, S. and Okuma, S., (1992) ”AdaptiveSliding Observers for Direct Field-Oriented Control of Induction Motor”, Proc.of 2nd IEEE Intl. Workshop on advanced motion control, 150-156, Nagoya,Japan.

18. Izosimov, D.B., (1983) ”Sliding Mode Nonlinear State Observer of an Induc-tion Motor”, in Meerov, M.V. and Kuznetsov, N.A. (Eds), Control of Multi-connected Systems (in Russian), 133-139, Nauka, Moscow.

19. Sabanovic, A., Jezernik, K. and Wada, K., (1996), ”Chattering Free SlidingModes in Robotic Manipulators Control”, Robotica, Vol.14, 17-29.

20. Morioka, H., Wada, K., Sabanovic, A. and Jezernik, K., (1995) ”Neural NetworkBased Chattering Free Sliding Mode Control”, Proceedings of SICE AnnualConference, 1303-1308, Japan.

21. Safaric, R., Hace, A. and Jezernik, K., (1994) ”Transputer Based TrajectoryTracking Neural Network Controller for a Robot Mechanism”, Proceedings ofthe IECON’94, 20th International Conference on Industrial Electronics, Controland Instrumentation, 794-799, Bologna, Italy.

22. Harashima, F., Hashimoto, H. and Kondo, S., (1985), ”MOSFET Converter-Fed Position Servo System with Sliding Mode Control”, IEEE Tran on IE, Vol.IE-32, No. 3.

23. Hashimoto, H., (1987) ”Variable Structure Strategy for Motion Control Sys-tems - Application to Electrical Machines”, Proc. of IECON’87, InternationalConference on Industrial Electronics, Control and Instrumentation, MA,USA.

24. Hashimoto, H., Yamamoto, H., Yanagisawa, S., and Harashim, F., (1988)”Brushless Servo Motor Control Using Variable Structue Approach”, IEEETran. on IA, Vol. 20, No. 1.

25. Hashimoto, H., Nakayama, T., Kondo, S., and Harashim, F., (1989) ”VariableStructue Approach for Brushless Servomotor Control”, IEEJ Tran. Vol. 109,No. 6.

26. Utkin, V., Guldner, J., and Shi, J., (1999) Sliding Mode Control in Electrome-chanical Systems, Taylor &Francis.

On the Development and Application ofSliding Mode Observers

Christopher Edwards1, Sarah K. Spurgeon1, and Chee Pin Tan1

Department of Engineering, University of Leicester, Leicester LE1 7RH, UK

1 Introduction

This chapter will provide a perspective on the development of sliding modeobservers for continuous time systems – primarily those which can be wellrepresented as linear systems subject to bounded nonlinearities/uncertainties.This encompasses a wide range of real engineering systems. Sliding mode ob-servers for systems modelled by nonlinear systems, and whose design method-ology explicitly exploits the nonlinear system structure, have recently beenconsidered in [1].

Consider initially a nominal linear system

x(t) = Ax(t) +Bu(t) (1a)y(t) = Cx(t) (1b)

where A ∈ IRn×n, B ∈ IRn×m and C ∈ IRp×n. Without loss of generalityassume that C has full row rank; i.e. there is no redundancy amongst themeasurements. The objective here is to obtain an estimate of the state x(t)by measuring the quantities y(t) and u(t). The design of a linear time in-variant dynamical system to estimate x(t) from the inputs and outputs wasextensively studied in the 1960’s by Luenberger. An algebraic condition onthe matrix pair (A,C) – the notion of observability – was proposed as a nec-essary and sufficient condition for state reconstruction [24]. (For simplicity,throughout this chapter, the observability condition will be assumed to hold,although technically some of the developments only require the weaker re-striction of detectability on the pair (A,C). See for example [3].) One way ofviewing the approach of Luenberger is to regard the observer as a model ofthe system in which the discrepancy between its output, and the output ofthe system, is fedback through a designer specified gain. Generally speaking,in sliding mode observers, instead of feeding back the output error betweenthe observer and the system linearly, the output error is fedback via a dis-continuous switched signal.

2 A Discontinuous Observer

To be specific, consider the dynamical system

z(t) = Az(t) +Bu(t) +Gnν (2)


where Gn ∈ IRn×p is to be specified, and ν is a discontinuous injection signaldepending on the output error ey(t) = Cz(t)− y(t).

As is common in sliding mode approaches, the analysis is greatly simplifiedif an appropriate change of coordinates is introduced. In the case of observerdesign, the coordinate transformation associated with the invertible matrix

Tc =[NT

c

C

](3)

where Nc ∈ IRn×(n−p) spans the null-space of C, is appropriate. As a resultof this transformation, the triple (A,B,C) has the form

A =[A11 A12

A21 A22

]B =

[B1

B2

]C =

[0 Ip

](4)

where the partitions of A and B are conformable with respect to C. Define thestate estimation error e(t) = z(t)− x(t) and suppose (in the new coordinatesystem) that the output error injection gain

Gn =[Gn,1

Ip

](5)

where Gn,1 ∈ IR(n−p)×p is a gain matrix to be specified.Suppose the state estimation error e(t) in the coordinates of (4) is parti-

tioned as (e1, ey) then

e1(t) = A11e1(t) +A12ey(t) +Gn,1ν (6a)ey(t) = A21e1(t) +A22ey(t) + ν (6b)

As argued in [32], if the components of the discontinuous term

νi = −ρ sign ey,i i = 1 . . .m (7)

where ey,i is the i-th component of ey, then for a large enough scalar ρ anideal sliding motion is induced in finite time on the surface

S = e ∈ IRn : ey = Ce = 0 (8)

During the sliding motion ey = ey = 0 and equation (6b) can be written as

0 = A21e1 + νeq (9)

where νeq represents the equivalent output error injection term necessary tomaintain a sliding motion on S. This is the natural analogue of the so-calledequivalent control occurring in the design of sliding mode controllers [32].Rearranging (9) and substituting back into equation (6a) it follows that thereduced order sliding motion is governed by

e1(t) = (A11 −Gn,1A21)e1(t) (10)

254 C. Edwards, S.K. Spurgeon, and C.P. Tan

In order to be able to sustain a sliding motion, and for (asymptotic) stateestimation error decay, the eigenvalues of (A11 − Gn,1A21) must be stable.However if (A,C) is observable then so is (A11, A21); see [32] for example.Therefore there exists Gn,1 for which (A11 −Gn,1A21) is stable.

Consider the example

A =[

0 1−1 0

]B =

[01

]C =

[1 1

](11)

An appropriate coordinate change to generate the canonical form in (4) isgiven by x → Tx where

T =[1 −11 1

](12)

This results in (with abuse of notation)

A =[

0 1−1 0

]B =

[−11

]C =

[0 1

](13)

In the following the gain Gn,1 = −1 which gives (A11 −Gn,1A21) = −1 andthe nonlinear gain ρ = 2. The sliding mode observer from (2) can then bewritten as

z1(t) = z2(t)− u(t) + 2sign ey(t) (14a)z2(t) = −z1(t) + u(t)− 2sign ey(t) (14b)

For simplicity assume that u(t) ≡ 0. In the simulations which follow theinitial condition for the observer (in the transformed coordinates) is [ 0 1 ]T

and the initial condition of the system (in the transformed coordinates) is[ 1 0 ]T. Figure 1 is a plot of the output estimation error ey against time. Thismay also be viewed as the switching function, and as expected, it is forcedto zero in finite time.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.5

1

Time, sec

Sw

itchi

ng fu

nctio

n

Fig. 1. Switching function

255On the Development and Application of Sliding Mode Observers

Figure 2 shows a comparison of the states of the system and the statesof the plant. The observer states are given by the solid line, and the plantstates by a dotted line. It is readily apparent that although ey = 0 in finitetime (after approximately 0.5 seconds) the state estimation error e(t) → 0exponentially and is different from zero for several seconds.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1

−0.5

0

0.5

1

1.5

1st s

tate

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

Time, sec

2nd

stat

e

Fig. 2. A comparison of the observer and system states

The description thus far describes the first published1 work on sliding modeobservers by Utkin [32].

An important consequence of inducing a sliding motion is that robust stateestimation can be obtained. In the same way that sliding mode controllersexhibit complete rejection of a class of uncertainty, so will the observer justdescribed. Suppose that the real system is governed by

x(t) = − sin x(t) (15)

and that the measured output corresponds to

y(t) = x(t) + x(t) (16)

1 It is more accurate to say the first published work in English on sliding observersis [32]. Earlier published work on this theme appears in Russian; see for examplethe references in [8].


Physically this may be viewed as a pendulum system where x(t) representsangular displacement from the vertical. The system in (11) may be viewedas a linearization of the system above at the equilibrium point (0, 0). Usingthe observer given in (14a)-(14b), the error system is governed by

e1(t) = ey(t)− sinx1(t)− ν (17a)ey(t) = −e1(t) + sinx1(t) + ν (17b)

where ν = −2 sign ey. During the sliding motion ey = ey = 0 and

νeq = e1(t)− sinx1(t) (18)

Substituting into equation (17a) the sine terms cancel leaving

e1(t) = −e1(t) (19)

The sliding motion is therefore independent of the nonlinearity/uncertaintyresulting from the linearization, and the observer states track the real statesasymptotically. Once again, in the simulations which follow the initial con-dition for the observer is given by [ 0 1 ]T and the initial condition of the(nonlinear) system is [ 1 0 ]T. Figure 3 is a plot of the output estimation errorey against time. Again the switching function is forced to zero in finite time.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.5

1

Time, sec

Sw

itchi

ng fu

nctio

n

Fig. 3. Switching function

Figure 4 shows a comparison of the states of the system and the statesof the plant. Once again perfect asymptotic tracking is obtained despite themismatch between the system and the linear model about which the observerwas designed.

3 Observers with Linear and Discontinuous Injection

The formulation just described generally requires large values of ρ in order toensure sliding for a broad range of initial state estimation errors – particularly


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1

−0.5

0

0.5

1

1.5

1st S

tate

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

Time, sec

2nd

stat

e

Fig. 4. A comparison of the observer and system states

if the underlying system is unstable. As discussed in [7], from a practicalviewpoint, this may cause difficulties. A trade-off is usually necessary betweenthe requirement of a large ρ to ensure a sliding mode occurs and its subsequentreduction to prevent excessive chattering (whilst still ensuring sliding). Forthis reason it may be preferable to include a linear output error injectionterm. Rather than the formulation described in (2) consider the dynamicalsystem

z(t) = Az(t) +Bu(t)−Gley(t) +Gnν (20)

where Gl, Gn ∈ IRn×p and ey(t) = Cz(t)−y(t). This is effectively the observerstructure of Slotine et al. [27], where it is argued that the linear gain shouldbe chosen to enhance the size of the so-called sliding patch i.e. the domain inthe state estimation error space in which sliding occurs. With a well designedlinear gain Gl this observer enjoys the same robustness properties. Indeed incertain situations (which will be discussed in more detail later) global stateestimation error convergence properties can be proven. To demonstrate this,consider once more the nonlinear system (15)-(16) and its linearization (11).Again change coordinates according to (12) to obtain the realization in (13).The observer design and analysis will be performed in this coordinate system.Choose as the output error injection feedback gains

Gl =[02

]Gn =

[−11

](21)


It can readily be shown that λ(A−GlC) = −1,−1. Furthermore the stateestimation error system is governed by

e1(t) = ey(t)− sinx1(t)− ν (22a)ey(t) = −e1(t)− 2ey(t) + sinx1(t) + ν (22b)

Consider now the quadratic form

V (e1, ey) = e21 + 2e1ey + 2e2

y = (e1 + ey)2 + e2y (23)

as a candidate Lyapunov function for the error system above. Clearly theexpression in (23) is positive definite. Furthermore, after some algebra, it canbe shown that

V = −2e21 − 6eye1 − 6e2

y + 2ey sinx1 + 2eyν

≤ −2e21 − 6eye1 − 6e2

y + 2|ey|| sinx1| − 2ρ|ey|≤ −2e2

1 − 6eye1 − 6e2y − |ey|

≤ −2(e1 + 32ey)2 − 3

2e2y − |ey| ≤ 0

and therefore (global) asymptotic stability of the error system has beenproven.

4 The Walcott and Zak Observer

The problem of robust state estimation for systems with bounded matcheduncertainty was first explored by Walcott & Zak [34]. They sought to designa sliding mode observer for the system

x(t) = Ax(t) +Bu(t) +Bf(t, y, u) (24a)y(t) = Cx(t) (24b)

where f : IR+ × IRp × IRm → IRm represents lumped uncertainty or nonlin-earities. The function is assumed to be unknown but bounded so that

‖f(t, y, u)‖ ≤ ρ(t, y, u) (25)

where ρ(·) is known. The observer proposed in [34] has the form

z(t) = Az(t) +Bu(t)−GCe(t)− P−1CTFTν (26)

where the discontinuous scaled unit vector term

ν = ρ(t, y, u) FCe(t)‖FCe(t)‖ (27)

and e(t) = z(t) − x(t). The symmetric positive definite matrix P ∈ IRn×n

and the gain matrix G are assumed to satisfy

PA0 +AT0 P < 0 (28)


where A0 := A−GC, and the structural constraint

PB = (FC)T (29)

for some F ∈ IRm×p. Under these circumstances the quadratic form given byV (e) = eTPe can be shown to guarantee quadratic stability [34]. Furthermorean ideal sliding motion takes place on

SF = e ∈ IRn : FCe = 0

in finite time.Remarks:

• If p > m then sliding on SF is not the same as sliding on Ce(t) = 0 andso the observer structure in (26) is different from the observer in (20).

• A system theoretic interpretation of (28)-(29) by Steinberg & Corless [29]is that the transfer function matrix G(s) = FC(sI − A0)−1B is strictlypositive real.

4.1 Synthesizing the Gains

The problem of synthesizing P,G and F (and incorporating some sort ofdesign element) is non-trivial. The original work in [34] postulated the use ofsymbolic manipulation tools to solve a sequence of constraints arising fromensuring that the principal minors of both P and the right hand-side of(28) are positive and negative respectively. In the original work it is also notapparent for which class of systems the design problem has a solution. For loworder systems the synthesis problem is quite tractable: the observer designedfor the pendulum described earlier is in fact a Walcott & Zak observer. Thestructural requirements (28)-(29) of Walcott & Zak were shown in [14] to besolvable if and only if

• rank(CB) = m• any invariant zeros of (A,B,C) ∈ C−

Details of the constructive design algorithms are described in [14] and followon from earlier work described in [9]. Under these circumstances a (semi)analytic expression for the solution to the observer design problem is givenin terms of a gain matrix L ∈ IR(n−p)×(p−m) and a stable matrix As

22 ∈ IRp×p.

5 A Linear Matrix Inequality Design Method

This section considers a more general problem than that of Walcott & Zakand proposes a numerically based solution methodology which exploits allthe degrees of freedom available in the design.


Consider the dynamical system

x(t) = Ax(t) +Bu(t) +Dξ(t, x, u) (30a)y(t) = Cx(t) (30b)

where A ∈ IRn×n, B ∈ IRn×m, C ∈ IRp×n and D ∈ IRn×q where p ≥ q.Assume C and D are full rank and the function ξ : IR+ × IRn × IRm → IRq

is unknown but bounded so that

‖ξ(t, x, u)‖ ≤ r1‖u‖+ α(t, y) (31)

where r1 is a known scalar and α : IR+ × IRp → IR+ is a known function.An observer of the form (as described earlier in §3)z(t) = Az(t) +Bu(t)−Gley(t) +Gnν (32)

will be considered whereGl ∈ IRn×p andGn ∈ IRn×p and ey(t) = Cz(t)−y(t).The discontinuous vector ν is defined by

ν =−ρ(t, y, u)‖PoD2‖ ey

‖ey‖ if ey = 00 otherwise

(33)

where Po ∈ IRp×p is symmetric positive definite. The matrices Po and D2 willbe defined formally later. The function ρ : IR+ × IRp × IRm → IR+ satisfies

ρ(t, y, u) ≥ r1‖u‖+ α(t, y) + γ0 (34)

where γ0 is a positive scalar. If the state estimation error e(t) := z(t)−x(t),then it is straightforward to show from equations (30a)-(30b) and (32) that

e(t) = A0e(t) +Gnν −Dξ(t, x, u) (35)

where A0 := A − GlC. In [14] it is argued that necessary and sufficientconditions for the existence of a stable sliding motion on

S = e ∈ IRn : ey = 0that is independent of ξ are

1. rank (CD)=q2. any invariant zeros of (A,D,C) lie in the left half plane

These of course are analogous to those for the Walcott & Zak observer de-scribed in §4.1. Again a (semi) analytic expression for the gains Gl and Gn interms of a matrix L ∈ IR(n−p)×(p−q) and a stable matrix As

22 ∈ IRp×p can bedemonstrated [14]. This parameterization however only represents a specificsubclass of possible solutions. Additional degrees of freedom are availablewhich are not exploited because the simplicity of solution is lost. The nextsection considers the design of the matrices Gl, Gn and Po so that a slidingmotion takes place on S. Assuming that conditions 1 and 2 are satisfied, anew parameterization is given which seeks to exploit all of the design freedomwhich is available.


5.1 A convex parameterization

A canonical form from [9] will constitute a useful starting point. If conditions1 and 2 above are satisfied there exists a coordinate transformation T0 inwhich the system (A,D,C) can be written as

A =

A11 A12

A211

A212A22

D =

[0D2

]C =

[0 T

](36)

where the sub-matrices A11 ∈ IR(n−p)×(n−p), A211 ∈ IR(p−q)×(p−n) representa detectable pair; D2 ∈ IRq×q is nonsingular; and T ∈ IRp×p is orthogo-nal. Further the unobservable modes of (A11, A211) are the invariant zerosof (A,D,C) [12]. In order to make the partitions in (36) conformable it isconvenient to introduce the term

D2 :=[0(p−q)×q

D2

](37)

Let Gl and Gn represent the observer gain matrices in the new coordinatesystem and define A0 = A − GlC. The gain matrix Gl is to be determinedbut assume

Gn =[−LTT

TT

]P−1

o (38)

where L ∈ IR(n−p)×p and L =[L 0

]with L ∈ IR(n−p)×(p−q); and the

orthogonal matrix T is part of the output distribution matrix C from (36).

Proposition 1. If there exists a positive definite Lyapunov matrix P , thatsatisfies P A0 + AT

0 P < 0, with the structure

P =[

P1 P1LLTP1 P2 + LTP1L

]> 0 (39)

where P1 ∈ IR(n−p)×(n−p) and P2 ∈ IRp×p, then the error system in equation(35) is quadratically stable.

Proof : Consider the quadratic form given by

V(e) = eTP e (40)

as a candidate Lyapunov function where e := T0e. Notice that if P1, P2 > 0then P > 0 from the Schur expansion. From (35) the derivative along thesystem trajectory

V = eT(AT0 P + P A0)e+ 2eTP Gnν − 2eTP Dξ (41)

From the definitions in (36), (38) and (39)

P Gn =[

0P2T

T

]P−1

o = CT (42)


if the symmetric positive definite matrix

Po := T P2TT (43)

Using the special structures of L and D2, LD2 = 0 and therefore

P D =[

0P2D2

]= CTPoD2 (44)

if the matrix

D2 := TD2 (45)

Consequently, (41) becomes

V = eT(AT0 P + P A0)e+ 2ey

Tν − 2eyTPoD2ξ

≤ eT(AT0 P + P A0)e− 2ρ‖PoD2‖‖ey‖ − 2ey

TPoD2ξ

Using the uncertainty bounds for ξ from equations (31) and (34)

V ≤ eT(AT0 P + P A0)e− 2ρ‖PoD2‖‖ey‖+ 2‖PoD2‖ [r1‖u‖+ α(y)] ‖ey‖

≤ eT(AT0 P + P A0)e− 2γ0‖PoD2‖‖ey‖

Since (AT0 P + P A0) < 0 it follows that V < 0 for all e = 0.

Corollary 1. An ideal sliding motion takes place on S in finite time. Fur-thermore the sliding dynamics are given by the system matrix A11 + LA211.

Proof : Using Proposition 1, a modification to Corollary 6.1 in [12] showsthat sliding takes place on S in finite time. Using the concept of equivalentoutput error injection, the sliding motion is governed by

(I − Gn(CGn)−1C)A0 =[A11 + LA211 A12 + LA22

0 0

]

Hence the sliding motion is governed by A11 + LA211 as claimed. Remarks: Since (A11, A211) is detectable by construction, there exists afamily of matrices L ∈ IR(n−p)×(p−q) such that A11 + LA211 is stable. If afurther linear change of co-ordinates

TL =[In−p L0 T

](46)

is applied to the triple (A, D, C) and its Lyapunov matrix P , the systemmatrix, disturbance distribution matrix and the output distribution matrixwill be in the form

A =[A11 A12

A21 A22

]D =

[0D2

]C =

[0 Ip

](47)


where A11 = A11 + LA211. In the new co-ordinate system, the Lyapunovmatrix will be

P = (TL−1)TP (TL

−1) =[P1 00 Po

](48)

where Po is defined in (43). The nonlinear output error injection gain matrix

Gn =[

0P−1

o

](49)

As argued in [9], the fact that P is a block diagonal Lyapunov matrix forA0 = A − GlC implies that A11 is stable and hence the sliding motion isstable.

The remainder of this chapter focuses on design methods to synthesise thegain Gl and the Lyapunov matrix P which has the structure given in (39).The problems will be posed in such a way that Linear Matrix Inequalities(LMIs) [2] can be used to numerically synthesise the required matrices.

5.2 Synthesis procedure for the gain matrices

In this section, P and Gl will be chosen so that the matrix inequality

AT0 P + P A0 < −PWP − P GlV GT

l P (50)

is satisfied, where the design weighting matrices W and V are assumed to besymmetric positive definite, and P has the structure in (39). Substituting forA0, the inequality (50) can be written as

ATP + P A− (Y C)T − Y C + PWP + Y V Y T < 0 (51)

where Y := P Gl. Using standard matrix manipulations, inequality (51) isidentical to

P A+ATP +(Y T−V −1C)TV (Y T−V −1C)−CTV −1C+ PWP < 0 (52)

Using inequality (52), the necessary and sufficient condition for (51) to holdis that P satisfies

P A+ ATP − CTV −1C + PWP < 0 (53)

since choosing

Y T = V −1C (54)

eliminates the third term in (52). The problem considered here is one ofminimizing trace(P−1) subject to P satisfying inequality (53). The observergain Gl can then be directly calculated as Gl = P−1CTV −1 which follows


from equation (54) and the definition of Y . The matrix inequality in (53) isequivalent to

[P A+ ATP − CTV −1C P

P −W−1

]< 0 (55)

by using the Schur complement. If X ∈ IRn×n is symmetric positive definitethen the LMI[−P I

I −X

]< 0 (56)

is equivalent to X > P−1. Thus minimizing trace(P−1) subject to (53) isequivalent to minimizing trace(X) subject to the LMIs (55) and (56). WritingP from (39) as

P =[P11 P12

PT12 P22

]> 0 (57)

where P11 ∈ IR(n−p)×(n−p), P22 ∈ IRp×p and

P12 :=[P121 0

](58)

with P121 ∈ IR(n−p)×(p−q), it follows there is a one-to-one correspondencebetween the variables (P11, P121, P22) and (P1, L, P2) since

P11 = P1 (59a)L = P−1

11 P121 (59b)P2 = P22 − PT

12P−111 P12 (59c)

The problem can then be formally defined as

Minimize trace(X) with respect to P11, P121, P22 and X subject to (55),(56) and (57).

This represents a convex optimization problem. Standard LMI software, suchas [16], can be employed to synthesise numerically P and X .

Remark: The motivation for the choice of the inequality posed in (50),and the minimisation of trace(P−1) subject to (53) and (56), is that in theabsence of uncertainty and as γ0 → 0, the observer behaves with sub-optimalLinear Quadratic Gaussian performance properties. For details see [31].

5.3 Design of the sliding motion system matrix

In the previous section, the dynamics of the sliding motion, although guaran-teed to be stable, were designed somewhat implicitly. This section considersthe sliding motion design problem and shows how additional LMI constraints


can be augmented with those used previously to tune the sliding mode per-formance. Using the new co-ordinates obtained after applying the transfor-mation TL in equation (46), the matrix inequality (53) can be written as

PA+ATP − CTV −1C + PWP < 0 (60)

where W := TLWTLT. The top left block of (60) can be identified to be

P1A11 +AT11P1 + P1W1P1 < 0 (61)

where W1 ∈ IR(n−p)×(n−p) > 0 is the top left sub-block of the matrix W andA11 = A11 + LA211. If the weighting matrix W is partitioned as

W =[W11 W12

WT12 W22

](62)

where W11 ∈ IR(n−p)×(n−p) and W12 is the null space of LT for all L, theninequality (61) can be written as

P1A11 +AT11P1 + P1W11P1 + P1LW22L

TP1 < 0 (63)

This is identical in structure to inequality (50) and hence W11 and W22

may be interpreted as playing the roles of performance and noise attenuationmatrices in a Linear Quadratic Gaussian sense for the observer problem as-sociated with the pair (A11, A211). Thus the choice of W11 and W22 can beused to tune the sliding motion. However since

P1A11 = P11A11 + P121A211 (64)

which is affine with respect to the LMI optimization variables P11 and P121,additional LMIs can be employed together with (55) and (56) to tune the slid-ing mode performance. One approach is to use root clustering [17] methodsto achieve pole placement of A11 in regions of the complex plane. Typicallythe poles may be required to lie in

• a conic sector centred at (0,0) with inner angle θa

• a disc of radius ra and centre (qa, 0)• a vertical strip aa < x < ba

Chilali and Gahinet [6] prove that the following inequalities describe theseregions[

(P1A11 +AT11P1) sin 1

2θa −(P1A11 −AT11P1) cos 1

2θa

(P1A11 −AT11P1) cos 1

2θa (P1A11 +AT11P1) sin 1

2θa

]< 0 (65a)

[ −raP1 P1A11 − qaP1

AT11P1 − qaP1 −raP1

]< 0 (65b)

[P1A11 +AT

11P1 − 2baP1 00 −(P1A11 +AT

11P1) + 2aaP1

]< 0 (65c)


Writing P1A11 as in (64), and substituting into (65a) - (65c), results in awell-posed convex problem since the inequalities (65a) - (65c) are affine inP11 and P121. A new optimization problem can be stated as

Minimize trace(X) with respect to P11, P121, P22 and X subject to theLMIs (56), (55), (57) and (65a) - (65c).

5.4 An example

The design method described in §5 will now be demonstrated by using a 7thorder aircraft example taken from [20]. The system matrices are

A =

0 0 1.0000 0 0 0 00 −0.1540 −0.0042 1.5400 0 −0.7440 −0.03200 0.2490 −1.0000 −5.2000 0 0.3370 −1.1200

0.0386 −0.9960 −0.0003 −2.1170 0 0.0200 00 0.5000 0 0 −4.0000 0 00 0 0 0 0 −20.0000 00 0 0 0 0 0 −25.0000

DT=

[0 0 0 0 0 20 00 0 0 0 0 0 25

]

C =

0 −0.1540 −0.0042 1.5400 0 −0.7440 −0.03200 0.2490 −1.0000 −5.2000 0 0.3370 −1.1200

1.0000 0 0 0 0 0 00 0 0 0 1.0000 0 0

where the states are respectively bank angle (rad), yaw rate (rad/s), roll rate(rad/s), sideslip angle (rad), washout filter state, rudder deflection (rad) andaileron deflection (rad). The inputs are rudder command(rad) and aileroncommand(rad). The measured outputs are roll acceleration (rads/s2), yawacceleration(rads/s2), bank angle (rad) and washout filter state. Here D ischosen to be the input distribution matrix.

The eigenvalues of the sliding motion represented by the system matrixA11 were required to lie in the intersection of the following regions:

• a circle of centre (0,0) and radius 5• a vertical upper bound at x = −2• a conic sector symmetric about the real axis, with inner angle θ = 80

The weighting matrices associated with the Ricatti-like inequality (53)were chosen respectively as W = diag(0.08, 0.08, 0.08, 0.08, 0.08, 0.05, .05) andV = 0.2 I2. The following matrices were obtained for the canonical form


described in (36):

A =

−2.0722 5.0994 1.6893 0 −0.1801 0.5527 −0.64650.0000 −0.0000 −0.0000 0 0.0000 −0.4607 −0.89690.0000 0.0000 0.0000 0 −0.0000 0.9884 −0.4625

−0.0000 0.4962 0.0226 −4.0000 0 0.0000 0.00000.0000 0.0122 −0.9159 0 0 0 0.0000

−20.1535 −3.2909 −6.9001 0 0.1583 −25.7219 0.625715.4751 5.0661 −1.3973 0 −0.1370 2.3305 −20.4769

DT=

[0 0 0 0 0 0 16.33530 0 0 0 0 25.8356 −10.8242

]

C =

0 0 0 0 0 −0.4126 −0.91090 0 0 0 0 −0.9109 0.41260 0 0 0 1.0000 0 00 0 0 1.0000 0 0 0

From this representation

A11 =

[−2.0722 5.0994 1.68930.0000 0.0000 0.00000.0000 0.0000 0.0000

]A211 =

[0.0000 0.4962 0.02260.0000 0.0122 −0.9159

]

The triple (A11, A211) has an unobservable mode and thus the system has aninvariant zero at -2.0722.

Following the synthesis procedure in §5.2 and imposing ,the design con-straints described earlier the following matrices were obtained:

L =

[−3.5281 −0.1519−1.6175 −0.27220.5549 0.8274

]and Po =

1.8468 0.4484 0.2924 3.38270.4484 0.2267 0.0900 0.47510.2924 0.0900 2.9843 −1.38173.3827 0.4751 −1.3817 84.7852

The associated gain matrices (in the original coordinates) are

Gl =

−0.3118 −0.1570 1.7300 0.0415−0.3436 0.2567 0.4682 0.1131−0.3279 −0.0215 1.3387 0.06620.1763 −0.1187 −0.2839 −0.0560

−0.1715 0.1856 0.0415 0.0655−7.4899 15.5815 −0.2850 0.08726.7480 −33.0829 0.2813 0.0864

Gn =

0 0 1.0000 00.0000 −0.0000 0.2327 1.5800

−0.0000 0.0000 0.7611 0.52800.0000 −0.0000 −0.1458 −0.7626

0 0 0 1.0000−1.3269 0.0379 −0.3518 −2.0294−0.3993 −0.8814 −0.0568 2.8100


The eigenvalues of A0 are

−70.3610,−23.6149,−4.0551− 0.7589± 0.9881i,−1.9097,−1.2967

and the eigenvalues of A11 are −2.0722,−0.5273,−1.0239

6 Fault Detection and Isolation

It has been seen that numerically tractable algorithms are available for thedesign of sliding observers. One obvious application for such observers is asestimators in state feedback control schemes [10,22]. Another, perhaps lessobvious, use is in fault detection and isolation (FDI) schemes. The funda-mental purpose of an FDI scheme is to generate an alarm when an abnormalcondition, such as a component malfunction or variation in operating condi-tion, develops in the process being monitored and to identify the source orlocation. Overviews and surveys of early work are given in [26,15,5]. Manyof the approaches proposed in the literature utilise Luenberger observers orKalman filters designed around mathematical models of the system to gener-ate estimates of the system states. Usually this additional information is usedto generate so-called residuals formed from the weighted difference betweenthe measured system outputs and the outputs from the observer. Differentapproaches are largely distinguished in the way in which the available designfreedom in the output error injection gain matrix is utilized and the way inwhich the residual information is processed. For example the gain matrix maybe designed to make the residuals sensitive to certain faults and not others,or else to make the residual vector lie in a specific direction in response toa particular fault. An overview of the current state of the art is detailed in[25] and the references therein. Here however the approach will be one offault estimation. Fault estimation provides a direct estimate of the size andseverity of the fault, which can be important in many applications [4,30]. Thesliding mode observers considered previously will be designed for FDI pur-poses. Their efficacy will be substantiated by considering their application tothe ship propulsion benchmark problem [23].Consider a nominal linear system subject to certain faults described by

x(t) = Ax(t) +Bu(t) +Dfi(t) (66a)y(t) = Cx(t) + fo(t) (66b)

where A ∈ IRn×n, B ∈ IRn×m, C ∈ IRp×n, D ∈ IRn×q with q ≤ p < n and thematrices C and D are of full rank. The functions fi(t) and fo(t) are deemedto represent actuator and sensor faults respectively and are assumed to beunknown but bounded. It is further assumed that the states of the systemare unknown and only the signals u(t) and y(t) are available. (Note: in thecase when p = n all the states will be known and the approach of Sreedhar[28] may be adopted.) The objective is to synthesise an observer to generate


a state estimate z(t) and output estimate Cz(t) such that a sliding mode isattained in which the output error

ey(t) = Cz(t)− y(t) (67)

is forced to zero in finite time. The observer structure (32) in §5 will be con-sidered. It will be shown that, provided a sliding motion can be attained,estimates of fi(t) and fo(t) can be computed from approximating the equiv-alent output error injection signal required to maintain a sliding motion.

6.1 Reconstruction of Actuator Faults

Consider initially the case when fo = 0. Assume the conditions necessaryfor the existence of a sliding mode observer are met for the triple (A,D,C).After first establishing the canonical form (36) from §5 and applying thecoordinate transformation associated with TL from (46), the state estimationerror system satisfies

e1(t) = A11e1(t) +A12e2(t)− Gl,1ey(t) (68a)e2(t) = A21e1(t) +A22e2(t)− Gl,2ey(t) + P−1

o ν −D2fi(t) (68b)

where (e1, e2) represents a partition of the state estimation error commensu-rate with the partition in (47) and the matrix A11 (which governs the slidingmotion) has stable eigenvalues. In this scenario ey = e2. During the slidingmotion, ey = 0 and ey = 0 and therefore equation (68b) becomes

0 = A21e1(t)−D2fi(t) + P−1o νeq (69)

where νeq is the equivalent output error injection signal. From (68a), andusing the fact that A11 is stable, it follows that e1(t) → 0 and therefore

P−1o νeq → D2fi(t) (70)

One way to recover the equivalent output error injection signal is by the use ofa low pass filter [32,12]. Here an alternative approach will be employed: sup-pose that the discontinuous component in (33) is replaced by the continuousapproximation

νδ = −ρ‖PoD2‖ ey

‖ey‖+ δ(71)

where δ is a small positive scalar. It can be shown that the equivalent outputerror injection signal can be approximated to any degree of accuracy by (71)for a small enough choice of δ. Since rank(D2) = q it follows from (70) that

fi(t) ≈ −ρ‖PoD2‖(DT2 D2)−1DT

2 P−1o

ey(t)‖ey(t)‖+ δ

(72)

The key point is that the signal on the right hand side of the above can becomputed on-line and depends only on the output estimation error ey.


6.2 Reconstruction of Sensor Faults

Now consider the case when fi(t) ≡ 0 and consider the effect of fo(t). In thissituation

ey(t) = e2(t)− fo(t) (73)

and it follows that

e1(t) = A11e1(t) +A12e2(t)− Gl,1ey(t) (74a)e2(t) = A21e1(t) +A22e2(t)− Gl,2ey(t) + P−1

o ν (74b)

or more conveniently

e1(t) = A11e1(t) +A12fo(t) + (A12 − Gl,1)ey (75a)

ey(t) = A21e1(t) +A22fo(t) + (A22 − Gl,2)ey − fo(t) + P−1o ν (75b)

Arguing as before, provided a sliding motion can be attained,

0 = A21e1 − fo(t) +A22fo(t) + P−1o νeq (76)

If the fault signal fo(t) is slowly varying compared with the dynamics of thesliding motion in (75a), e1 ≈ −A−1

11 A12fo and the derivative fo in (76) canbe ignored. Consequently

P−1o νeq ≈ −(A22 −A21A−1

11 A12)fo (77)

As in the previous section, the continuous approximation to the equivalentoutput injection νeq can be calculated from (71) and if (A22 −A21A−1

11 A12)is nonsingular the fault signal can be obtained from equation (77). Note, if(A22 − A21A−1

11 A12) is singular, inference can still be made about certainfault channels depending on the precise nature of the rank deficiency [14].

The remainder of the chapter considers the application of this approachto the ship propulsion system benchmark proposed in [23].

7 An FDI Scheme for the Ship Benchmark

This section considers the use of a sliding mode FDI scheme applied to aship propulsion system benchmark problem [23]. The benchmark consists ofa propulsion system for a low speed marine vehicle. The consequences of faultsin such a system have serious safety implications as well as significant lossesassociated with capital investment. The propulsion system consists of twoidentical engine/propeller systems which are placed in parallel. A continuoustime state-space description of one engine is

x(t) = Ax(t) +Bu(t) + Ed(t) + F1f(t) + ws(t) (78a)y(t) = Cx(t) + F2f(t) + w(t) (78b)


where d(t) represents a vector of unknown disturbances, the reference/controldemand vector

u(t) =[θref

Y

]pitch angle referencefuel index (79)

and f(t) is a vector of faults. The outputs represent

y(t) =

θ

nU

propeller pitch angleshaft speedship speed

(80)

and w(t) and ws(t) are noise vectors. In particular,

ws(t) =[−kt w1(t) 0 0 0

]T (81)

where the constant kt = 0.1510 and w1(t) is the first component of w(t). Thephysical meaning of the components of the unknown fault signal f(t) is givenbelow:

f(t) =

f1(t)f2(t)f3(t)

Pitch angle measurement fault

Hydraulic leakage (incipient) faultShaft angular measurement fault

(82)

In addition, as in [23], a multiplicative fault corresponding to a failure asso-ciated with the diesel engine governor will be considered so that

u(t) =[

θref (t)Y (t) +∆kyY (t)

](83)

where ∆ky represents an unknown change in gain.A linearization of the nonlinear model from [23] is given by the triple

A =

−0.1510 0 0 0−2.2936 −0.3154 0.1964 8.52520.1424 0.0056 −0.0149 0

0 0 0 −1.8745

B =

0.1500 0

0 00 00 1.0000

C =

1 0 0 00 1 0 00 0 1 0

(84a)

The fault distribution structure in the framework described earlier can beexpressed as

fo(t)T :=[f1(t) f3(t) 0

](85)

where f1(t) and f3(t) are components of f(t) as in (82). The actuator faultdistribution matrix is

D :=

1 00 4.5480 00 0

and fi(t) :=

[f2(t)− ktf1(t)

f∆kyY (t)

](86)

Details of this derivation are given in [13].


7.1 Observer design

This system (A,D,C) has a stable invariant zero at −1.8745 and in thecanonical form of (36) the triple (A,D, C) is given by

A =

−1.8745 0 0 0

0 −0.1510 0 08.5252 −2.2936 −0.3154 0.1964

0 0.1424 0.0056 −0.0149

D =

0 01 00 4.5480 0

(87a)

C =

0 1 0 00 0 1 00 0 0 1

(87b)

Choosing W = diag0.3, 6.0, 6.0, 1.0 and V = diag0.0018, 0.0132, 0.0100yields observer gain matrices (in the original coordinates) as

Gl =

23.4158 −0.1584 0.0128−1.1613 21.7424 0.10450.0712 0.0792 24.48000.0839 1.8379 −0.0067

Gn =

0.0421 −0.0021 0.0001−0.0021 0.2784 0.00110.0001 0.0011 0.2448

0 0 0

and

Po =

23.7352 0.1829 −0.0132

0.1829 3.5939 −0.0158−0.0132 −0.0158 4.0851

7.2 A sliding mode fault detection system

From (70), and writing νeq := P−1o νeq for notational convenience, the actuator

faults satisfy[νeq,1

νeq,2

]=

[1 00 4.548

] [fi,1

fi,2

](88)

It can readily be verified that

−(A22 −A21A−111 A12) = −A22 =

0.1510 0 0

2.2936 0.3154 −0.1964−0.1424 −0.0056 0.0149

(89)

since A12 = 0. Note: this means that equation (76) no longer depends one1(t). From equation (77)[

νeq,2

νeq,3

]=

[2.2936 0.3154

−0.1424 −0.0056

] [fo,1

fo,2

](90)

or equivalently, since these equations furnish a unique solution[fo,1

fo,2

]=

[−0.1740 −9.82354.4366 71.4434

] [νeq,2

νeq,3

](91)


Define[fest,1

fest,2

]:=

[0 −0.1740 −9.82350 4.4366 71.4434

]νeq (92)

so that the signals νeq,1, νeq,2 and fest,1 and fest,2 can be used to reconstructdifferent fault scenarios. Note four distinct fault events need to be isolatedfrom only three available equivalent output error injection signals. To be ableto distinguish between actuator faults and sensor faults additional informa-tion needs to be exploited. One possibility is to consider the output error (orswitching function) signal ey since step changes in sensor faults will causea break in the sliding motion. This can be seen from equation (73) since atthe instant a step change occurs in fo, ey momentarily becomes equal to thefault before the nonlinear output error injection term quickly forces ey backto zero. Thus sensor faults can be detected by monitoring ey for brief breaksin the sliding motion and the nature of the fault can be deduced from fest,1

and fest,2. Conversely actuator faults tend not to interrupt the sliding motionand manifest themselves in terms of changes in νeq,1 and νeq,2.

The following remarks can be made:

• an actuator fault f2, corresponding to an incipient hydraulic/pitch ratefault, affects only fi,1 and hence in turn (from equation (88) and (92))only affects νeq,1 and not fest.

• a sensor fault f3 corresponding to a change in shaft angular velocityaffects only fo,2 and thus from equations (91) and (92) can be detectedby a peak in the second component of ey and replicated from the secondcomponent of fest.

• a fault f1 corresponding to a change in pitch affects both fi,1 (fromequation (86)) and also fo,1 although it can be seen from equations (91)and (92) that replication of the fault is only made through fest,1. Atthe instant at which a fault occurs a peak will be observed in the firstcomponent of ey.

• an actuator fault relating to a change in the controller gain ∆kyY will bereplicated by νeq,2 (from equation (88)) but will also cause a variation infest as shown in equation (92). As argued earlier, this type of fault willnot interrupt the sliding motion and therefore can be distinguished fromsensor faults.

In the simulations which follow ρ = 20 and δ = 0.001. The fault eventshave been simulated as shown in Table 1.

In Figure 5 six signals are displayed and used to deduce both the oc-currence and the exact nature of any faults. Specifically the signals plottedrepresent the difference between νeq,1, νeq,2, fest,1 and fest,2 for both enginesand the ‖ey‖ signals. For clarity, the initial portion of the time histories of

2 The incipient hydraulic fault as modelled in Figure 3 [23] is argued to result in aslow drift in pitch angle - hence the symbol ∆θ in the notation of [23].


0 500 1000 1500 2000 2500 3000 3500−0.05

0

0.05Di

ffere

nce (

theta

dot)

Time

0 500 1000 1500 2000 2500 3000 3500−5

0

5

Diffe

renc

e (de

lta k)

Time

0 500 1000 1500 2000 2500 3000 3500−1

−0.5

0

0.5

1

Diffe

renc

e (the

ta)

Time

0 500 1000 1500 2000 2500 3000 3500−20

−10

0

10

20

Diffe

renc

e (n)

Time

0 500 1000 1500 2000 2500 3000 3500−0.5

0

0.5

Norm

of S

witch

ing F

uncti

on 1

0 500 1000 1500 2000 2500 3000 3500−0.5

0

0.5

Norm

of S

witch

ing F

uncti

on 2

Time

Fig. 5. Switching function and fault reconstruction signals


Event Notation Start End Type Size

Pitch angle measurement fault (∆θ) f1 180 210 step −1.00Hydraulic leakage incipient fault1 (∆θ) f2 800 1700 ramp 0.01Pitch angle measurement fault (∆θ) f1 1890 1920 step 0.64Shaft angular velocity measurement fault (∆n) f3 680 710 step −4.30Shaft angular velocity measurement fault (∆n) f3 2640 2670 step −13.70Gain drop in diesel engine (∆ky) ∆ky 3000 3500 step 1.00

Table 1. Fault events for the linear simulation

the reconstruction signals are given in Figure 6. After 180 seconds the peakin the first component of the switching function ey1,1 indicates the occur-rence of a pitch angle sensor fault. Examining fest,1 and fest,2 it can be seenthat both react (Figure 6). In addition the pitch rate/hydraulic leakage re-construction signal νeq,1 has also reacted strongly. It transpires that on thenonlinear simulation a pitch fault affects the shaft angular velocity and viceversa. However, a pitch fault causes the pitch rate estimator to react; thisdoes not happen for faults relating to the angular velocity measurement ofthe shaft speed. Also the ‘step input’ nature of the fault is modified by thefact the corrupted pitch angle measurement signal is fed-back and regulatedby the controller. The result is the upward/downward ‘double hit’ response.

A peak in the second component of the switching function ey1,2 occursafter 680 seconds, indicating the presence of a shaft speed sensor fault. Exam-ining fest,1 and fest,2 in Figure 6 it can be seen that both react. Importantlyno significant change occurs in the pitch rate estimator νeq,1 which confirmsthat there has been a fault associated with the measurement of the angularvelocity of the shaft. (Using similar logic, the switching function peaks at1890 seconds and 2640 seconds in Figure 5 indicate pitch and shaft speedfaults respectively.)

No further peaks occur in the switching functions. However in Figure 7(at approximately 800 seconds), it can be seen that a drift starts to occur inthe hydraulic leakage/pitch rate estimate. This suggests an actuator fault -specifically one associated with hydraulic leakage/pitch rate - which can beseen to induce a drift in both the pitch and shaft indicator signals. A recoveryis made at approximately 1700 seconds and the fault estimation signals returnto near zero.

From Figure 5, after 3000 seconds, a step change occurs in both the fest

signals (and in the νeq,2 signal). This is not heralded by a peak in the switch-ing function signals indicating the cause is an actuator fault. Since the hy-draulic leakage/pitch rate estimator does not react, νeq,2 indicates the pres-ence of a control gain change. An appropriate threshold needs to be set basedon the results of a fault free simulation.

It is readily seen that the outputs from the sliding mode based FDI ob-server have enabled all the faults present in the test sequence to be identifieddespite the presence of model uncertainty, load changes and noise.


0 100 200 300 400 500 600 700 800 900 1000−0.05

0

0.05Di

ffere

nce

(thet

a do

t)

Time

0 100 200 300 400 500 600 700 800 900 1000−2

−1

0

1

2

Diffe

renc

e (de

lta k)

Time

0 100 200 300 400 500 600 700 800 900 1000−0.5

0

0.5

Diffe

renc

e (th

eta)

Time

0 100 200 300 400 500 600 700 800 900 1000−10

−5

0

5

10

Diffe

renc

e (n)

Time

Fig. 6. Fault reconstruction signals (magnified)

0 500 1000 1500 2000 2500 3000 3500−0.01

−0.005

0

0.005

0.01

Diffe

renc

e (th

eta

dot)

Time

Fig. 7. Fault reconstruction signals (magnified)


The plots presented in Figures 5-7 are associated with the simulationsspecified in the benchmark [23]. However it is possible to modify the setupto remove the noise which is introduced on the three ‘measured’ signals andproduce ‘ideal’ reconstructions. In this way it is possible to explore the effectsof noise on the reconstruction signals. In the following simulations the faultscenarios are those described in Table 1. Figures 8 and 9 correspond to Figures6 and 7; the only difference is the simulations associated with Figures 8 and 9are noise free. The solid lines represent the ideal reconstruction signals fromthe noise free simulation, the dotted lines are the data in Figures 6 and 7 forcomparison. It can be seen that the reconstruction signal associated with thebenchmark setup represent a ‘noisy’ version of the ideal reconstructions.

0 100 200 300 400 500 600 700 800 900 1000−0.05

0

0.05

Diffe

renc

e (the

ta do

t)

Time

0 100 200 300 400 500 600 700 800 900 1000−2

−1

0

1

2

Diffe

renc

e (de

lta k)

Time

0 100 200 300 400 500 600 700 800 900 1000−0.5

0

0.5

Diffe

renc

e (the

ta)

Time

0 100 200 300 400 500 600 700 800 900 1000−10

−5

0

5

10

Diffe

renc

e (n)

Time

Fig. 8. Comparisons between the benchmark and noise free reconstructions


0 500 1000 1500 2000 2500 3000 3500−0.01

−0.005

0

0.005

0.01Di

ffere

nce (

theta

dot)

Time

Fig. 9. Comparisons between the benchmark and noise free reconstructions

It should be noted that in the benchmark setup the noise has zero meanand appears to have no low frequency affect on the reconstruction signals.

8 Summary

This chapter has discussed the development of sliding mode observers for aclass of uncertain systems. Recent work has been reported which uses Linearmatrix Inequalities to formulate the design problem in convex terms so thatefficient numerical methods can be employed to synthesise the observer gains.A case study has also been presented which demonstrates how sliding modeobservers can be used for fault Detection and Isolation.

9 Notes and References

The pendulum example is effectively the case study considered in [33]. Inter-estingly in this paper a sliding mode observer was compared to other directnonlinear observer design methodologies for the pendulum example. It wasargued that the sliding mode observer had the best performance and was theleast involved in terms of design.

An almost completely analytic solution to the Walcott & Zak observerproblem is given in [11] for square systems. In [10] an analysis is given ofthe closed loop system arising from using a state feedback sliding mode con-troller implemented using the state estimates generated from a sliding modeobserver. For a class of systems with matched uncertainty the closed loopretains the invariance properties of its full state feedback counterpart. Anoverview of much of this work appears in [12].

In Herrmann et al. [22] the minimum phase condition is alleviated tosome extent for systems with many outputs. In essence, rather than seekinga closed-loop observer based controller which is robust to matched uncer-tainty, robustness with respect to a strict subset of matched uncertainty isachieved by effectively generating an uncertainty distribution matrix of lowerdimension which is contained within the range space of the input distributionmatrix. For details see [22].


The use of sliding mode observers for disturbance estimation is discussedin [19]. In fact this work describes a design framework for a broader class ofsystems than the one described in this chapter – namely linear time varyingsystems.

Sliding mode ideas have previously been used in the literature for faultdetection. Sreedhar et al. [28] consider a model-based sliding mode observerapproach although in their design procedure it is assumed that the statesof the system are available; a different approach is adopted by Hermans &Zarrop [21] who attempt to design an observer in such a way that in thepresence of a fault the sliding motion is destroyed. The approach in [14] seeksto design the sliding mode observer to maintain sliding even in the presence offaults. More detailed simulations and analysis of a (slightly different) slidingmode based FDI scheme is described in [13].

An interesting use of the concept of equivalent output error injection isdescribed in [18]. A recursive design procedure based on successive evaluationsof appropriate output error injection signals is shown to provide estimationof all unknown states of observable linear systems in finite time.

10 Acknowledgements

The first author would like to acknowledge the support of the University ofLeicester in granting him study leave, during which time this chapter waswritten.

References

1. Barbot, J.P., (2001) Sliding mode observers, In Perruquetti W., Barbot J.P.,editors, Sliding Modes in Automatic Control, Marcel Dekker.

2. Boyd S.P., El Ghaoui L., Feron E., Balakrishnan V. (1994) Linear Matrix In-equalities in Systems and Control Theory, SIAM: Philadelphia.

3. Brogan W.L., (1991) Modern Control Theory, Prentice Hall, Englewood CliffsNJ.

4. Chen J., Patton R.J. (1993) Fault estimation in linear dynamic systems, InProceedings of the 12th IFAC World Congress, 7:501–504.

5. Chen J., Patton R.J. (1999) Robust Model-Based Fault Diagnosis for DynamicSystems, Kluwer Academic Publsihers.

6. Chilali M., Gahinet P. (1996) H∞ design with pole placement constraints: anLMI approach, IEEE Transactions on Automatic Control, 41, 358–367.

7. Dorling C.M., Zinober A.S.I. (1983) A comparative study of the sensitivity ofobservers, In Proceedings of the IASTED Symposium on Applied Control andIdentification–Copenhagen, 6.32–6.38.

8. Drakunov S., Utkin V.I. (1995) Sliding mode observers: tutorial, In Proceedingsof the 34th IEEE Conference on Decision and Control, 3376–3378.

9. Edwards C., Spurgeon S.K. (1994) On the development of discontinuous ob-servers, International Journal of Control, 59, 1211–1229.


10. Edwards C., Spurgeon S.K. (1996) Robust output tracking using a sliding modecontroller/observer scheme, International Journal of Control, 64 967–983.

11. Edwards C., Spurgeon S.K. (1997) Sliding mode output tracking with applica-tion to a multivariable high temperature furnace problem, International Jour-nal of Robust and Nonlinear Control, 7, 337–351.

12. Edwards C., Spurgeon S.K. (1998) Sliding Mode Control: Theory and Applica-tions, Taylor & Francis.

13. Edwards C., Spurgeon S.K. (2000) A sliding mode control observer based FDIscheme for the ship benchmark, European Journal of Control, 6 341–356.

14. Edwards C., Spurgeon S.K., Patton R.J. (2000) Sliding mode observers for faultdetection, Automatica, 36, 541–553.

15. Frank P.M. (1990) Fault diagnosis in dynamic systems using analytical andknowledge based redundancy - a survey and some new results, Automatica, 26,459–474.

16. Gahinet P., Nemirovski A., Laub A., Chilali M. (1995) LMI Control Toolbox,User Guide, MathWorks, Inc.

17. Gutman S., Jury E. (1981) A general theory for matrix root-clustering in sub-regions of the complex plane, IEEE Transactions on Automatic Control, 26853–863.

18. Hasakara I., Ozguner U., Utkin V.I. (1998) On sliding mode observers viaequivalent control approach, International Journal of Control, 71 1051–1067.

19. Hashimoto H., Utkin V.I., Xu J.X., Suzuki H., Harashima F. (1990) VSS ob-servers for linear time varying systems, In Proceedings of the 16th Annual Con-ference of the IEEE Industrial Electronic Society, 34–39.

20. Heck B.S., Yallapragada S.V., Fan M.K.H. (1995) Numerical methods to designthe reaching phase of output feedback variable structure control, Automatica,31 275–279.

21. Hermans F.J.J., Zarrop M.B. (1996) Sliding mode observers for robust sensormonitoring, Proceedings of the 13th IFAC World Congress, 211–216.

22. Herrmann G., Spurgeon S.K., Edwards C. (2001) A robust sliding mode outputtracking control for a class of relative degree zero and non-minimum phaseplants: a chemical process application, to appear in International Journal ofControl.

23. Izadi-Zamanabadi R., Blanke M. (1997) A ship propulsion system as a bench-mark for fault-tolerant control, In Proceedings of the IFAC Symposium - Safe-process ’97, 1074 – 1081.

24. Luenberger D.G. (1971) An introduction to observers, IEEE Transactions onAutomatic Control, 16 596–602.

25. Niemann H. (2000) Editorial, special issue on fault detection and identification,International Journal of Robust and Nonlinear Control, 10 1153–1154.

26. Patton R.J., Frank P.M., Clark R.N. (1989) Fault Diagnosis in Dynamic Sys-tems: Theory and Application, Prentice Hall, New York.

27. Slotine J.J.E., Hedrick J.K., Misawa E.A. (1987) On sliding observers for non-linear systems, Transactions of the ASME: Journal of Dynamic Systems, Mea-surement and Control, 109 245–252.

28. Sreedhar R., Fernandez B., Masada G.Y. (1993) Robust fault detection in non-linear systems using sliding mode observers, In Proceedings of the IEEE Con-ference on Control Applications, 715–721.

29. Steinberg A., Corless M.J. (1985) Output feedback stabilisation of uncertaindynamical systems, IEEE Transactions on Automatic Control, 30 1025–1027.


30. Stoustrup J., Grimble M.J. (1997) Integrating control and fault diagnosis: aseparation result, In Proceedings IFAC Symposium, Safeprocess ’97, Hull, 323–328.

31. Tan C.P., Edwards C. (2000) An LMI approach for designing sliding modeobservers, In Proceedings of the IEEE Conference of Decision and Control,Sydney.

32. Utkin V.I. (1992) Sliding Modes in Control Optimization, Springer-Verlag,Berlin.

33. Walcott B.L., Corless M.J., Zak S.H. (1987) Comparative study of nonlinearstate observation techniques, International Journal of Control, 45 2109–2132.

34. Walcott B.L., Zak S.H. (1987) State observation of nonlinear uncertain dynam-ical systems, IEEE Transactions on Automatic Control, 32, 166–170.


Multivariable Output-FeedbackSliding Mode Control

Liu Hsu1, Jose Paulo Vilela Soares da Cunha2, Ramon R. Costa1, andFernando Lizarralde3

1 COPPE/Federal University of Rio de Janeiro, Brazil2 State University of Rio de Janeiro, Brazil3 Dept. of Electronic Eng./Federal University of Rio de Janeiro, Brazil

Abstract. This chapter addresses the problem of model-reference sliding modecontrol of uncertain linear multi-input-multi-output systems by output-feedback.After a brief revision of the main approaches and results available in the literature,a new method to solve the problem is presented using unit vectors as switching func-tions to induce sliding modes. It is named Unit Vector Model-Reference AdaptiveControl (UV-MRAC).

Some features of the UV-MRAC are: (i) No explicit state observers are em-ployed. (ii) It applies to plants of any uniform relative degree. (iii) Less restrictiveapriori knowledge about the high frequency gain matrix of the plant is required forits implementation, namely, only a Hurwitz condition is required.

The analysis of the algorithm shows that, in spite of possible bounded inputdisturbances, the model following error equation is exponentially stable with respectto some small residual set. The peaking phenomena, which is a problem in mostobserver based variable structure controllers, does not occur in the UV-MRAC.

1 Introduction

In real world applications, the model of a plant is often uncertain with re-spect to the system parameters, nonlinearities and disturbances. In order toguarantee stability and good performance of a control system even in thepresence of significant uncertainty, robust and/or adaptive control schemeshave been widely investigated in the last two decades.

A large number of works have used the Variable Structure Control (VSC)approach with sliding modes [39,8]. Under ideal conditions, this techniquecan guarantee invariant system performance once sliding mode is attained.For this reason, some authors consider VSC as an adaptive control method[20,17,6,7]. In this sense, the term adaptive is used in the present work. How-ever, adaptation in this case is based on signal synthesis rather than param-eter estimation.

A close approach to VSC is the nonlinear robust control design basedon the Lyapunov method [16,15] which relies on unit vector control (UVC)instead of the usual sign switching control of VSC. The common feature ofthese approaches is that they are directly or indirectly based on sliding modes.


Hence, in what follows, we will use the term sliding mode control (SMC) torefer either to one or the other.

The sliding mode is designed to occur on some manifold of the statespace expressed by σ(x) = 0, where σ : R

n → Rm is some smooth function.

To convey the main idea in a simple way, consider a linear plant given by

x = Ax + Bu , y = Cx , (1)

x ∈ Rn, u, y ∈ R

m, and a plane given by σ(x) = Sx = 0. A necessarycondition for the sliding mode to occur on Sx = 0 is rank(SB) = m, i.e.,full rank. In other words, we say that the relative degree n∗ of the transferfunction from u to σ must be one. This necessary condition is not a difficultyif the plant states are available. However, when only the plant output y ismeasurable, the condition applies also to the matrix Kp = CB which mustalso be full rank, i.e., the plant transfer function must have relative degreeone. Unfortunately, in many practical applications, this condition does nothold and the design of output-feedback VSC becomes a difficult problem,particularly when the plant is uncertain. One way to solve the problem is touse state observers.

1.1 State Observers in VSC

Owing to the practical difficulty of measuring all states, output-feedbackstrategies for VSC and SMC were developed [30,33]. In this respect, a seminalpaper is [4] where asymptotic observers were introduced in VSC to addressa fundamental problem of SMC, namely, the realization of “ideal” slidingmodes in practice. The key idea of [4] (see also [38]) is to design the slidingmodes in the observed state (x). The sliding manifold is given in terms ofσ(x) = Sx, which has time derivative

σ = SAx + SBu + SKC(x− x) . (2)

Now, the rank condition for the existence of the sliding mode applies to SB,i.e., S must be chosen so that SB has full rank. Note that σ is the auxiliaryerror which can be calculated for any S since x is totally known.

Then, ideal sliding is possible using asymptotic state observers, indepen-dently of the original relative degree of the plant transfer function Gyu(s).

It is also worthwhile noting that Gyu(s) is not required to be minimumphase. This requirement applies only to Gσu(s) since the sliding motion isgoverned by the transmission zeros of Gσu(s) [43].

The main point is that the observer based system allows the existence ofa linear transfer function with n∗ = 1 and stable transmission zeros closing aloop, called the ideal sliding loop (ISL), around the SMC block (discontinuouscontrol), independently of the plant relative degree, as shown in Fig.1. Theexistence of an ISL is necessary for the realization of ideal sliding modes.

284 L. Hsu et al.

S

Observer

Plant

y=Cx

−

+y=Cx

x

σ

ISL

SMC

Fig. 1. Sliding mode control using a state observer

1.2 High Gain Observers for Uncertain Systems

Full knowledge of the plant is assumed in [4]. To guarantee robustness to plantuncertainties, sliding observers or high gain observers were used [34,40,10][11,30,28].

Usually the goal is robust stabilization by output-feedback, e.g., [40,44,30].A simple illustrative example of SMC stabilization using high gain observerscan be found in [30]. In this approach, the inverse of the observer gain ischaracterized by a small parameter ε and as typical result the controllerwill be guaranteed to work only if there exists a constant ε∗ > 0 such that0 < ε < ε∗. For example, consider the plant

x1 = x2 , x2 = u + d(x1) , y = x1 . (3)

Then, regarding d(x1) as a disturbance, a high gain observer could be [30]

˙x1 = x2 +1ε(y − x1) , ˙x2 = u +

1ε2

(y − x1) . (4)

Besides the problems with noise sensitivity inherent to high-gain systems, theundesirable phenomena of peaking in the observer and controller signals canarise [11,30].

1.3 Model-Reference Approach for Sliding Mode Control

A more general problem is to specify the desired closed loop response bya reference model which is to be followed by the plant. This approach wasintroduced in the VSC literature in [42] where state feedback was utilizedand full state model following was required. It was further developed in[45]. Output-feedback VSC for output tracking was considered in [1,3] forsingle-input-single-output (SISO) systems. In these references, the structureof Model-Reference Adaptive Controllers (MRAC) by output-feedback (as in

285Multivariable Output-Feedback Sliding Mode Control

[29]) was introduced in the VSC theory. Following this approach, new con-trollers were proposed in [20,17,19] for SISO plants. An early reference onoutput-feedback adaptive control of multi-input-multi-output (MIMO) sys-tems is [37]. It should be noted that in the latter, SMC was used as anauxiliary control signal to achieve robustness and disturbance rejection. TheMIMO case was considered in [6], [7].

In this chapter, an output-feedback based unit vector MRAC (UV-MRAC)for MIMO plants is presented. While the controller of [35] uses a state spacedescription of the plant and a nonlinear observer, the approach here, likewise[37,6,7], relies on the plant transfer matrix and follows the development ofMRAC without explicit state observers [9,36]. The controller presented hereis an extension of the scheme developed in [20,17,19,23] for SISO systems.The controller guarantees global exponential stability with respect to somesmall residual set in the error space which can be made arbitrarily small de-pending on a design parameter. The required apriori knowledge about thehigh frequency gain (HFG) matrix is less restrictive than in previous worksand no peaking phenomena [11] occurs in the closed loop system.

2 Preliminaries

2.1 Norms and notations

The following notation and basic concepts are employed in this chapter.

• The maximum and the minimum singular values of a matrix A are de-noted as σmax(A) and σmin(A), respectively.

• The maximum and minimum eigenvalues of a symmetric matrix P aredenoted as λmax(P ) and λmin(P ), respectively.

• ‖x‖ denotes the Euclidean norm of a vector x and ‖A‖ = σmax(A) denotesthe corresponding induced norm of a matrix A.

• The L∞e norm of the signal x(t) ∈ Rn is defined as [26]

‖xt‖∞ := sup0≤τ≤t

‖x(τ)‖ . (5)

• Mixed time-domain and Laplace transform domain (operator) represen-tations will be adopted. However, to give a precise meaning for such repre-sentations, the following notation and concepts are adopted. The outputy of a linear time-invariant system with transfer function H(s) and inputu is given by H(s)u. Pure convolution operations h(t) ∗ u(t), h(t) beingthe impulse response from H(s), will be eventually written, for simplicity,as H(s)∗u. Consider the realization x = Ax+Bu, y = Cx+Du, of H(s).Then,

y(t) = H(s)u(t) = h(t) ∗ u(t) + C exp(At)x(0) , (6)

where the exponential term is the homogeneous response of the system(u(t) ≡ 0).

286 L. Hsu et al.

• The symbol s denotes either the complex variable of Laplace transformsor the differential operator d

dt in time-domain expressions.• The norm of an operator H(s) is defined as

‖H(s)‖ := ‖h(t)‖1 =∫ +∞

0

‖h(τ)‖dτ . (7)

The following inequality holds for y(t) = h(t) ∗ u(t) ([26], p. 80)

‖yt‖∞ ≤ ‖h(t)‖1 ‖ut‖∞ . (8)

• A stability margin δ of a Hurwitz polynomial p(λ) is defined as

0 < δ ≤ mini

Re(−λi) , (9)

where λi are the roots of p(λ). Similarly, for a Hurwitz matrix A or aHurwitz transfer function G(s), a stability margin is defined with λi beingthe eigenvalues of A or the poles of G(s).

2.2 Basic MIMO Systems Concepts

Let A,B,C be a realization of a strictly proper and nonsingular m × mrational transfer function matrix G(s) = C(sI −A)−1B.

• The observability index of the pair C,A (A ∈ Rn×n, C ∈ R

m×n) is thesmallest integer ν, (1 ≤ ν ≤ n), such that

Oν =[CT (AC)T · · · (Aν−1C)T

]T(10)

has full rank. The observability index has a nice system interpretation:(ν − 1) is the largest number of derivatives of y required to determinex(0) [27] (pp. 356–357). In other words, it gives information about theorder of the state variable filters required in the structure of a MIMOMRAC design.

• Another important figure in a MIMO MRAC design is the relative de-gree of the plant, which requires the concept of interactor matrix for itsdefinition. The interactor matrix associated to G(s) is a special lowertriangular matrix with the form [41]

ξ(s) = H(s) diagsn

∗1 , sn

∗2 , · · · , sn

∗m

, (11)

H(s) =

1 0 . . . 0h21(s) 1 . . . 0

......

...hm1(s) hm2(s) . . . 1

, (12)


where the polynomials hij(s) are either zero or divisible by s, such that

Kp = lims→∞ ξ(s)G(s) (13)

is finite and nonsingular. Notice that ξ(s) defined in (11) is unique and itis shown in [41] that ξ(s) is invariant under dynamic compensation. Thematrix Kp ∈ R

m×m in (13) is called high frequency gain (HFG) matrixand (n∗

1, n∗2, · · · , n∗

m) is the vector relative degree of G(s). If n∗i = n∗,

(i = 1, · · · ,m), we say that G(s) has uniform vector relative degree n∗.

2.3 Discontinuous Differential Equations

• Since we shall deal with discontinuous systems, it is necessary to defineprecisely the meaning of a solution for such systems. Here, Filippov’sdefinition is assumed [12]. Note that the control signal u is not necessar-ily a function of t in the usual sense when sliding modes take place. Inorder to avoid clutter, we will denote by u(t) the locally integrable func-tions which are equivalent to u, in the sense of equivalent control, alongany given Filippov solution z(t) of the closed-loop system. It should bestressed that z(t) is, by definition, absolutely continuous. This definitionis motivated by the adequate representation of the behavior of physicalsystems when the actual switching mechanism tends to an ideal switchingdevice which corresponds to the given discontinuous differential equation[13, pp. 94–99] [39]. Then, along any such solution, u can be replacedby u(t) in the right-hand side of the governing differential equations.Although the equivalent control u(t) = ueq(t) is not directly available,for affine systems filtering u with any strictly proper filter G(s) givesG(s)u = G(s)u(t) = G(s)ueq(t).

• The extended equivalent control is defined as an equivalent control whichapplies for a complete system motion, i.e., on and outside the slidingmode surface σ(x(t), t) = 0. Consider the class of nonlinear systems de-scribed by x = f(x, t) +B(x, t)u , where f(·) and B(·) are smooth vectorfields. Let x(t) be a solution of this system, for t ∈ [0, T ). Then, the ex-tended equivalent control is a locally integrable function, defined almosteverywhere in the interval [0, T ), and is given by

ueq = −[GB(x(t), t)]−1[Gf(x(t), t) +d

dtσ(x(t), t)] , (14)

where G = ∂∂xσ(x(t), t). The above expression is well defined since the

solution x(t) is absolutely continuous by definition and, thus, has deriva-tives almost everywhere.

288 L. Hsu et al.

3 Problem Statement

3.1 Plant Description

We consider an observable and controllable MIMO linear time-invariant plantdescribed by

xp = Apxp + Bp[u + d(t)] , (15)y = Cpxp , (16)

where xp ∈ Rn is the state, u ∈ R

m is the input, d ∈ Rm is an unmeasurable

input disturbance, and y ∈ Rm is the output. The corresponding input-output

model is

y = G(s)[u + d(t)] , (17)

where G(s) = Cp(sI − Ap)−1Bp is an m × m transfer function matrix. Weassume that the parameters of the plant model are uncertain, i.e., only knownwithin certain finite bounds.

3.2 Basic Assumptions

The following assumptions regarding the plant are taken as granted:

(A1) The transmission zeros of G(s) have negative real parts.(A2) G(s) has full rank and is strictly proper.(A3) The observability index ν of G(s) is known.(A4) The interactor matrix ξ(s) is diagonal and G(s) has known uniform

vector relative degree n∗.(A5) A matrix Sp is known such that −KpSp is Hurwitz.(A6) The disturbance d(t) is piecewise continuous and a bound d(t) is known

such that ‖d(t)‖ ≤ d(t) ≤ dsup < +∞, ∀t ≥ 0.

Assumptions (A1) to (A3) are usual in MIMO adaptive control. Someprior knowledge of the interactor is usually assumed in MIMO adaptive con-trol and also VSC literature [36,37,6,7]. However, if uncertain systems are tobe considered, some assumptions may not be acceptable in practice. In thissense, assumption (A4), also made in [21], may look too strong. However, itcan be argued that a diagonal interactor can be achieved with an appropriateprecompensator. Indeed, in most cases (in a generic sense), Lemma 2.6 in [36]guarantees that there exists a precompensator Wp(s) so that G(s)Wp(s) hasdiagonal interactor matrix (see also [14,32]). Moreover, Wp(s) does not de-pend on the parameters of G(s). Once the interactor is known to be diagonaland if the relative degree of each element of G(s) (or of G(s)Wp(s)) is known,then ξ(s) can be determined without any prior knowledge about the transferfunction parameters [41]. In order to achieve uniform vector relative degree,


one can follow the approach of [7] which employs a precompensator to renderthe relative degree uniform and equal to n∗ = maxi n∗

i .Assumption (A5) represents a considerable extension of the class of ac-

ceptable high frequency gain matrices. In other published works the morerestrictive assumption of positive definiteness of KpSp (and also symmetryin some approaches) is required [26,36,37,6,7].

3.3 Reference Model

The reference model is defined by

yM = WM (s) r , r, yM ∈ Rm , (18)

where, without loss of generality,

WM (s) = diag

1s + γj

L−1(s) , γj > 0 , (j = 1, · · · ,m) , (19)

L(s) = L1(s)L2(s) · · ·LN (s) , N = n∗ − 1 , (20)

Li(s) = (s + αi) , αi > 0 , (i = 1, · · · , N) , (21)

The reference signal r(t) is assumed piecewise continuous and uniformlybounded. WM (s) has the same uniform vector relative degree n∗ as G(s) andits high frequency gain is the identity matrix (i.e., lims→∞ ξ(s)WM (s) = I).The tracking of more general reference models could be obtained by simplypreshaping the reference signal r through a precompensator at the input ofthe above model.

3.4 Control Objective

The objective is to design a control law u so that the plant output y asymp-totically tracks the output yM of the reference model, within some smallresidual error. More precisely, the control objective is to achieve asymptoticconvergence of the output error

e(t) = y(t) − yM (t) (22)

to zero, or to a small residual neighborhood of zero in the error space, ast → +∞.

4 Unit Vector Control

The unit vector control law has the form

u = −-(x, t)v(x)

‖v(x)‖ , (23)

290 L. Hsu et al.

where x is the state vector, v(x) is a vector function of the (partial) state ofthe system (e.g., the output) and -(x, t) ≥ 0, ∀x, ∀t. We refer to -(·) as theunit vector modulation function, which is designed to induce a sliding modeon the manifold v(x) = 0. We will henceforth assume that u = 0 if v(x) = 0only to have a complete definition of the control law.

4.1 Basic Lemmas

Some lemmas regarding the application of the unit vector control into theMRAC framework are now introduced. These lemmas generalize their SISOcounterparts found in [23] and are instrumental for the controller synthesisand stability analysis.

In what follows, we assume t∈R+ so that ∀t means ∀t≥0, except other-

wise stated. We use “LI” to denote locally integrable in the sense of Lebesgueand omit the term “almost everywhere” since its need is believed obviouswhere necessary. We denote by π(t) any exponentially decreasing signal, i.e.,‖π(t)‖ ≤ Re−λt, ∀t, for some unknown positive scalars R and λ.

Proposition 1. Consider the MIMO system

ε(t) = Aε(t) + K[u + d(t) + π(t)] , (24)

where A,K∈Rm×m, d(t) and π(t) are LI. Assume that −K is Hurwitz. If

u = −-(ε, t)ε

‖ε‖ , -(ε, t) ≥ δ + cε‖ε(t)‖ + (1 + cd)‖d(t)‖ , (25)

where - is LI, cε ≥ 0 and cd ≥ 0 are appropriate constants, and δ ≥ 0 is anarbitrary constant, then, for the closed loop system (24)–(25), the inequality

‖ε(t)‖ ≤ (c1‖ε(0)‖ + c2R) e−λ1t (26)

holds ∀t for some positive constants c1, c2 and λ1. Therefore, the system isglobally exponentially stable when π(t) ≡ 0. Moreover, if δ > 0, then thesliding mode at the point ε = 0 is reached after some finite time ts ≥ 0.

Proof. see Appendix B.1.

Lemma 1. Consider the MIMO system

ε(t) = M(s)[u + d(t) + π(t)] , (27)

where M(s) is a minimum phase (m × m) transfer function matrix withuniform vector relative degree one and high frequency gain matrix K, −Kbeing Hurwitz, and d(t) and π(t) are LI. If

u = −-(ε, t)ε

‖ε‖ , -(ε, t) ≥ δ + cε‖ε‖ + (1 + cd)‖d(t)‖ + cfεf , (28)


where - is LI, δ ≥ 0 is an arbitrary constant, cf , γ0, cε, cd are appropriatenonnegative constants, and εf is generated by the filter

εf = −γ0εf + γ0‖ε‖ , (29)

then, the inequality

‖ε(t)‖ and ‖xea(t)‖ ≤ [c1‖xea(0)‖ + c2R] e−λ1t (30)

holds ∀t≥0 for some positive constants c1, c2, λ1, where xTea = [xTe εf ] is thecomplete state of the closed loop system and xe is the state of any stabilizableand detectable realization of (27) (possibly nonminimal). Moreover, if δ > 0,then ε(t) becomes identically zero after some finite time ts ≥ 0.


A more specialized version of Lemma 1 is the following corollary.

Corollary 1 (Lemma 1). Suppose, in Lemma 1, that M(s) = diag1/(s+αi)K with −K Hurwitz. Then all the properties of Lemma 1 hold for

-(ε, t) ≥ δ + cε‖ε‖ + (1 + cd)‖d(t)‖ , ∀t . (31)

Moreover, if αi = α > 0, (∀i), then cε = 0.Proof. see Appendix B.3.

Lemma 2. Consider the MIMO system

˙ε(t) = −αε(t) + K[u + d(t)] , ε(t) = ε(t) + π(t) + β(t) , (32)

which has input-output relationship given by

ε(t) = L−1(s)K[u + d(t)] + π(t) + β(t) , (33)

where L(s) = (s + α)I, I is the m × m identity matrix, ε, ε, u, d ∈ Rm,

α> 0, −K ∈Rm×m is Hurwitz, d(t) is LI, and π(t) and β(t) are absolutely

continuous, (∀t). If u=−-(t) ε‖ε‖ , where - is LI and -(t) ≥ (1+ cd)‖d(t)‖, ∀t,

for some appropriate cd≥0, then, the signals ε(t) and ε(t) are bounded by

‖ε(t)‖ and ‖ε(t)‖ ≤ c1‖ε(0)‖e−αt + c2

[Re−min(α,λ)t + ‖βt‖∞

], (34)

for some positive constants c1, c2.Proof. see Appendix B.4.

4.2 Unit Vector Versus Variable Structure Control

Traditional VSC systems are based on the sign function (x ∈ Rm)

sgn(x) :=

sgn(x1)...

sgn(xm)

, sgn(xi) :=

1 if xi > 0,0 if xi = 0,−1 if xi < 0.

(35)

292 L. Hsu et al.

To compare the stability properties of VSC and UVC, let us consider theVSC system given by

x = Ksgn(x) , K =[k11 k12

k21 k22

], x ∈ R

2 . (36)

The following Theorem was proved in [22].

Theorem 1. Consider the discontinuous system (36). Then x(t) = 0 is aglobally asymptotically stable solution if and only if one of the following con-ditions holds:

(1) k12k21 = 0, k11 < 0 and k22 < 0.(2) k12k21 = 0, k11

|k12| + k22|k21| < 0, as well as det(K) > 0.

Equivalent conditions are also given in [13] (pp. 256–258). Unfortunately,no necessary and sufficient conditions are known for systems of dimensiongreater than two. In contrast, the following Theorem for UVC systems ofarbitrary dimension was proved in [2].

Theorem 2. Consider the system

x = K(x)x

‖x‖ , (37)

where x ∈ Rm, m ≥ 1, K : R

m → Rm×m, and det(K(x)) = 0, ∀x. The origin

of the state-space of system (37), with bounded K(x) and its derivatives, isstable (asymptotically stable, unstable) if and only if the system z = K(z)zis stable (asymptotically stable, unstable).

In particular, if K(x) is a constant matrix, we conclude that the originof the UVC system (37) is globally asymptotically stable if and only if K isHurwitz. It is straightforward to find matrices K which result in asymptoticstability with both control laws, e.g., K = −I. However there exist matriceswhich result in stable closed-loop systems with one control law only.

Another clear difference between VSC and UVC is that sliding modes inthe former may take place in any individual switching surfaces before reachingtheir intersection x = 0, whereas, for UVC, the sliding mode occurs only atthe origin.

4.3 Relationship with Adaptive Stabilization

A connection of UVC with adaptive stabilizers of [5,24] is now discussed.Consider the plant (15)–(16) with d(t) ≡ 0 under the same assumptions ofminimum phase and uniform vector relative degree one as in Lemma 1. Onecan apply the Byrnes-Willems adaptive stabilizer

u(t) = −k(t)y(t) , k(t) = ‖y(t)‖2 , k(0) ∈ R , (38)


where k : R+ → R is a scalar adaptive gain. Such control law leads to an

asymptotically stable output y if −Kp is Hurwitz [5].The design of the UVC requires the knowledge of bounds for the plant

parameters, which is not needed in adaptive stabilization. An advantage ofUVC is that exponential stability and even finite time convergence of the plantoutput can be achieved, whereas the adaptive stabilizers can only achieveasymptotic [5] and exponential convergence [24].

5 Control Parameterization for Output-Feedback

If the plant is perfectly known, then a control law which achieves matchingbetween the closed-loop transfer matrix and WM (s) is given by the followingparameterization, which appears in the adaptive control literature

u∗ = θ∗Tω −Wd(s) ∗ d(t) , (39)

where the parameter matrix θ∗ and the regressor vector ω(t) are given by

θ∗ =[θ∗T1 θ∗T2 θ∗T3 θ∗T4

]T, θ∗ ∈ R

2mν×m , (40)

ω =[ωT

1 ωT2 yT rT

]T, ω ∈ R

2mν , (41)

ω1 =B(s)Λ(s)

u , ω2 =B(s)Λ(s)

y , ω1, ω2 ∈ Rm(ν−1) , (42)

B(s) =[Isν−2 Isν−3 · · · Is I

]T, (43)

Wd(s) = I − θ∗T1

B(s)Λ(s)

, (44)

θ∗3 , θ∗4 ∈R

m×m and Λ(s) is a monic Hurwitz polynomial of degree ν−1. Thematching condition requires that θ∗T4 = K−1

p . Let X = [xTp ωT1 ωT

2 ]T . Theopen-loop system composed by the plant (15)–(16) and the filters (42) canbe written as

X = AoX + Bou + Bodd , (45)y = CoX . (46)

Then, the regressor vector is given by

ω = Ω1X + Ω2r , Ω1 =

0 I 00 0 ICp 0 00 0 0

, Ω2 =

000I

. (47)

Substituting u by θ∗Tω in (45) and omitting d, we obtain the followingnonminimal realization of WM (s)

XM = AcXM + Bcr , (48)yM = CoXM , (49)

294 L. Hsu et al.

where Ac = Ao+Boθ∗TΩ1, Bc = Boθ

∗TΩ2 = Boθ∗T4 . Note that Ac is Hurwitz.

The state space error equation is obtained by subtracting (48)–(49) from(45)–(46) resulting (Xe := X −XM )

Xe = AcXe + Bc(θ∗T4 )−1[u− θ∗Tω + Wd(s) ∗ d(t)

], (50)

e = CoXe , (51)

or in input-output form

e = WM (s)Kp

[u− θ∗Tω + Wd(s) ∗ d(t)

]. (52)

From the error equations, it becomes clear that if u is replaced by theideal matching control law u∗ (39), then e(t) → 0 exponentially as t → +∞.

From the control parameterization described above, we now make thefollowing assumption on the class of admissible control laws.

(A7) The control law satisfies the inequality

‖ut‖∞ ≤ Kω‖ωt‖∞ + Krd , (53)

where Kω, Krd are positive constants.

This assumption guarantees that no finite time escape occurs in the sys-tem signals. Indeed, in this case the system signals will be regular and there-fore can grow at most exponentially [31].

6 Design and Analysis of the UV-MRAC

This section considers the MRAC problem when the plant states are not fullyavailable and only output-feedback is possible. The solution described herestems from the variable structure model-reference adaptive controller (VS-MRAC) structure developed for SISO plants in [20,17,19] and generalizedto the MIMO case in [6,7]. The novelty is the use of unit vector controlinstead of the sign function of VSC and therefore the controller is referredto as UV-MRAC. Compared to the results of [6,7], the main new featuresare: (a) global exponential stability properties can be demonstrated, (b) lessrestrictive assumption on the plant high frequency gain matrix is required,and (c) the controller is shown to be free of peaking.

6.1 The Case of Relative Degree One

For n∗ = 1, we have N = n∗ − 1 = 0 and L(s) = I. Therefore, from theoutput error equations (50)–(52), and according to Corollary 1, the proposedunit vector control law is

u = unom − Sp-e

‖e‖ , unom = θnomTω , (54)


where Sp ∈ Rm×m is a design matrix which verifies assumption (A5) and

θnom is some nominal value for θ∗.From Corollary 1, exponential stability is achieved if the modulation signal

- ∈ R+ satisfies the inequality:

- ≥ δ + cε‖e‖ + (1 + cd)∥∥S−1

p

[(θnomT − θ∗T

)ω + Wd(s) ∗ d(t)

]∥∥ , (55)

where cε ≥ 0, cd ≥ 0 are appropriate constants and δ ≥ 0 is an arbitraryconstant. From Lemma 4, one possible choice for - which satisfies (55) moduloexponentially decaying terms is:

- = δ + c1 ‖ω‖ + c2 ‖e‖ + c3 d(t) , (56)

with

d(t) = d(t) +c4

s + λd∗ d(t)

(≥∥∥∥∥(I − θ∗T1

B(s)Λ(s)

)∗ d(t)

∥∥∥∥)

, (57)

where ci ≥ 0 (i = 1, . . . , 4) are appropriate constants, λd is the stabilitymargin of Λ(s) and d(t) ≥ ‖d(t)‖. It should be noted that the constants c1,c2 and c3 can be estimated based on some nominal parameters of the plant.The constant c4 should be such that the inequality in (57) holds.

Now, in order to state the stability theorem for this case, consider theerror system (50)–(51) and the augmented state vector zT = [XT

e , d0], where

d0 is the transient state of the filter (57). Then we can state the followingstability result.

Theorem 3. The UV-MRAC strategy (54), (56)–(57) for plants of uniformvector relative degree is globally exponentially stable, i.e. ‖z(t)‖ ≤ ke−λt‖z(0)‖(k, λ > 0), ∀t. Moreover, if δ > 0 in the modulation function (55), the outputerror e(t) becomes zero after some finite time.

Proof. The proof is a simple application of Corollary 1 to the nonminimalrealization of (52) given by the error system (50)–(51) and the equationsfor the transient state of the filter that generates d. The transient state isincorporated to the π term of Lemma 1.

6.2 The Case of Higher Relative Degree

For higher uniform relative degree the unit vector control strategy cannotbe applied directly. Similarly to the SISO case, to overcome this difficulty,[19,23], the controller structure is now modified according to Figs. 2 and 3. Akey idea for the controller generalization is the introduction of the predictionerror [19]

e = WM (s)L(s)Knom(U0 − L−1(s)UN

), (58)

296 L. Hsu et al.

Fig. 2. UV-MRAC for plants of higher uniform relative degree. The state filters andthe computation of 0 are omitted to avoid clutter. The realization of the operatorL is presented in Fig. 3

Fig. 3. Unit vector implementation of the operator L

where Knom is a nominal value of K = KpSp and the operator L(s) isgiven by (20). The purpose of L(s) is to make G(s)L(s) and WM (s)L(s) ofuniform vector relative degree one. The operator L(s) is noncausal but canbe approximated by the unit vector lead filter L presented in Fig. 3. The caseof relative degree one is recovered by letting L(s) = L = I. Then, e ≡ 0 andthus internal prediction error loop is eliminated resulting in the scheme ofSect. 6.1.

The averaging filters F−1i (τis) in Fig. 3 are low-pass filters with transfer

function given by

F−1i (τis) =

1fav(τis)

, (59)


with fav(τis) being Hurwitz polynomials in τis such that the filter has unitDC gain (fav(0) = 1), e.g., fav(τis) = τis+1. If the time constants τi > 0 aresufficiently small, the averaging filters give an approximation of the equivalentcontrol signals [39]

(Ui−1)eq ≈ F−1i (τis) Ui−1 . (60)

In [17], this approximation was used to justify the system stability properties.A complete theoretical justification of the VS-MRAC, taking into account theaveraging filter dynamics, was presented later in [19,23].

Simplified Analysis

A simplified analysis is helpful in understanding of the UV-MRAC. As in[17], assume that the modulation functions -i are such that ideal slidingmodes start after some finite time tsi in each loop around the unit vectorblocks. Then, neglecting the time constant τ of the averaging filters we have(Ui−1)eq = F−1

i (τis) (Ui−1) and thus (UN )eq = L(s)(U0)eq. Then, the inputand output of the block WMLKnom would be just exponentially decayingsignals. However, since sliding takes place in all unit vector loops, ε0 = 0 andhence, the output error e(t) must decrease to zero exponentially fast. In thecase of higher relative degree the error convergence is asymptotic. Finite timetransient is no longer possible due to the dynamics of the block WMLKnom.

This result is a guideline for the complete stability analysis in the sensethat it represents the limiting situation when the averaged controls tend tothe corresponding equivalent controls as the time constants τi → +0.

The role of the prediction error (e) loop is not evident from the simpli-fied analysis. However, without the prediction error loop there would be nopossibility of having an ideal sliding mode around the first unit vector blockcorresponding to U0 [18] due to the small lags introduced by averaging filters.Hence, chattering would be unavoidable. The main purpose of the predictionerror loop is to create the necessary ideal sliding loop around the first unitvector block.

Error Equations

We develop the expressions for the auxiliary error signals which are conve-nient for the controller design and stability analysis.

From (52) and (58), the auxiliary error signal ε0 = e− e can be rewrittenas

ε0 = WM (s)Kp

[u− θ∗Tω + Wd(s) ∗ d(t)

]−−WM (s)L(s)Knom

[U0 − L−1(s)UN

]. (61)

Using u = θnomTω − SpUN , K = KpSp from assumption (A5) and

U := (Knom)−1Kp

[(θ∗T − θnomT

)ω −Wd(s) ∗ d(t)

]−− (

I − (Knom)−1K)UN , (62)

298 L. Hsu et al.

the auxiliary error ε0 in (61) can be rewritten as

ε0 = WM (s)L(s)Knom[−U0 − L−1(s)U

], (63)

where it is clear that in the nominal case (θ∗ = θnom and K = Knom) thecontrol signal U0 has to reject only the effect of the disturbance d(t). Theauxiliary errors in the lead filters are given by

εi = F−1i (τs)Ui−1 − L−1

i (s)Ui , (64)

where, for the sake of simplicity, it is assumed that all the averaging filterstime constants are the same (τi ≡ τ). Applying U0 (obtained from (63)) in(64), the auxiliary error in the first lead filter is written as

ε1 = L−11 (s)

−F−11 (τs)L1(s)L−1(s)

[(Knom)−1W−1

M (s)ε0 + U]− U1

.

Through the recursive application of this procedure, the auxiliary errors canbe rewritten as (i = 1, . . . , N − 1):

εi = L−1i (s)

[−Ui − F−1

1,i (τs)L−1i+1,N (s)U

]− πei − π0i , (65)

εN = −L−1N (s)(Knom)−1K

[UN +F−1

1,N (τs)Ud

]−

− (I − (Knom)−1K

)βuN − πeN − π0N , (66)

where Li,j(s)=∏j

k=i Lk(s) (Li,j(s)=1 if j <i), Fi,j(τs) is defined in similarway and (by convention, πe1≡0)

Ud = S−1p

[(θ∗T − θnomT

)ω −Wd(s) ∗ d(t)

], (67)

βuN = (F1,N (τs) − I)F−11,N (τs)L−1

N (s)UN , (68)

πei =i−1∑j=1

Lj,i−1(s)F−1j+1,i(τs) εj = Li−1(s)F−1

i (τs)[πe,i−1+εi−1] , (69)

π0i = (WM (s)F1,i(τs)Li,N (s)Knom)−1ε0 . (70)

Bounds for the Auxiliary Errors

The error system to be considered here is composed of (50)–(51), (63), (65),and (66). Let Xε denote the state vector of (63) and x0

FL denote the transientstate corresponding to the following operators: L−1 in (63), F−1

1,i L−1i+1,N in

(65) and all the remaining operators associated with βuN , πei, π0i in (68)–(70) and di in (79), (81). Since all these operators are stable, there existpositive constants KFL and aFL such that

‖x0FL(t)‖ ≤ KFLe

−aFLt‖x0FL(0)‖ .

In order to fully account for the initial conditions, the following statevector z is used

zT = [(z0)T , εTN ,XTe ] , (z0)T = [XT

ε , εT1 , ε

T2 , . . . , ε

TN−1, (x

0FL)T ] . (71)


In what follows, all K’s and a’s denote positive constants, operator norms(‖·‖) are L∞ induced norms, and “Π” and “Π0” denote any term of the formK‖z(0)‖e−at and K‖z0(0)‖e−at, respectively, where K and a are (generic)positive constants.

Theorem 4. For N ≥ 1 consider the auxiliary errors (63), (65) and (66). If−Knom and −(Knom)−1K are Hurwitz, and the relay modulation functionssatisfy

-0 ≥ (1 + cd0)‖L−1 ∗ U‖ + cε0‖ε0‖ ,-i ≥ (1 + cdi)‖(F−1

1,i L−1i+1,N ) ∗ (U)‖ , (i = 1, · · · , N − 1) , (72)

-N ≥ (1 + cdN )‖F−11,N ∗ Ud‖ ,

for all t ≥ 0, with some appropriate constants cε0 ≥ 0 and cdi ≥ 0 fori = 0, . . . , N , then the auxiliary errors εi, (i = 0, · · · , N − 1), tend to zero atleast exponentially. Moreover,

‖εi(t)‖, ‖Xε(t)‖ ≤ Π0 , (73)‖εN (t)‖ ≤ τ

∥∥I − (Knom)−1K∥∥KeNC(t) + Π , (74)

and

‖πei(t)‖, ‖π0i(t)‖ ≤ Π0; i = 1, . . . , N , (75)‖βuN (t)‖ ≤ τKβNC(t) + Π0 , (76)

where

C(t) = Mθ‖ωt‖∞ + Mred , (77)

with some positive constants Mθ and Mred.


Remark 1. Note that in the above theorem the Hurwitz condition on −Knom

and −(Knom)−1K could be satisfied choosing Knom = knomI, with knom ∈R, knom > 0. In particular, with Knom = knomI the HFG condition is simply−K Hurwitz.

Error System Stability

The following stability theorem will be demonstrated for the full error systemgiven by (63), (65) and (66). For N = 0 (n∗ = 1) exponential stability followsfrom Theorem 3.

Theorem 5. For N ≥ 1 assume that −Knom and −(Knom)−1K are Hur-witz, and that the modulation functions satisfy (72). Then, for sufficientlysmall τ > 0, the error system (50), (63), (65) and (66) with state z as de-fined in (71) is globally exponentially stable with respect to a residual set oforder τ , i.e., there exist positive constants a and Kz such that ∀z(0), ∀t ≥ 0,‖z(t)‖ ≤ Kze

−at‖z(0)‖ + O(τ).


300 L. Hsu et al.

7 Implementation of the UV-MRAC

In this section we address the problem of obtaining explicit implementablemodulation functions for the UV-MRAC. We also present a very simplifiedversion which can be used in practice when only local stability is sufficient.

7.1 Modulation Functions

For the case n∗ = 1, explicit modulation functions were obtained in the form(56–57). We now develop similar expressions for the case of n∗ > 1. Forthe design of -i, i = 0, . . . , N , we apply Lemma 4 to the inequalities (72)obtaining, for -0,

-0 = δ0 + cε0‖ε0‖ + cω0‖L−1(s)ω‖ + cU0‖L−1(s)UN‖ + d0(t) , (78)

d0(t) = cd0d(t) +cd0

s + λd0∗ d(t) , (79)

with appropriate constants cε0, cω0, cU0, cd0, cd0 > 0, arbitrary δ0 ≥ 0, andλd0 being the stability margin of the filter L−1(s)Wd(s).

Similarly, for -i, i = 1, . . . , N − 1, we obtain

-i = δi + cωi

∥∥∥F−11,i (τs)L−1

i+1,N (s)ω∥∥∥+

+ cUi

∥∥∥F−11,i (τs)L−1

i+1,N (s)UN

∥∥∥+ di(t) , (80)

di(t) = cdid(t) +cdi

s + λdi∗ d(t) , (81)

with arbitrary δi ≥ 0 and appropriate constants cωi, cUi, cdi, cdi > 0, andλdi is a stability margin of the filter F−1

1,i (τs)L−1i+1,N (s)Wd(s). For -N we have,

-N = δN + cωN

∥∥∥F−11,N (τs)ω

∥∥∥+ dN (t) , (82)

with arbitrary δN ≥ 0 and appropriate constants cωN , cdN , cdN > 0, dN (t)given by (81) and λdN is a stability margin of the filter F−1

1,N (τs)Wd(s).

7.2 Simplified Modulation Functions

One can further simplify the modulation - by using Lemma 4 to compute asimple upper bound for ‖ω‖ which does not involve the utilization of a largeamount of filtering devices that would be needed for the direct computationof ω. To this end notice that, by virtue of Lemma 4, one can write (moduloexponentially decaying terms)

‖ω1‖ ≤ c1s + λ

‖y‖ ,

where λ is the stability margin of Λ(s). Similarly, one could find an upperbound for ‖ω2‖ ≤ c2/(s + λ)‖u‖. However, since u involves discontinuous


terms, this would result in a conservative estimate [23]. For this reason wedevelop another bound using the following identity which holds for any λ > 0and τ > 0:

1s + λ

=τ

τs + 1+

1 − τλ

(τs + 1)(s + λ). (84)

Then, it follows that∥∥∥∥ 1s + λ

u

∥∥∥∥ ≤ τ‖uav‖ +cλ

s + λ‖uav‖ , uav =

u

τs + 1, (85)

where we see that the latter bound does not involve ‖ueq(t)‖ but rather theaveraged control uav(t). This, in principle, leads to less conservative (smaller)modulation functions.

Then, we have the following upper bound for ‖ω‖:

‖ω‖ ≤ c1s + λ

‖y‖ + τcτ‖uav‖ +cλ1

s + λ‖uav‖ + ‖y‖ + ‖r‖ . (86)

This bound can be used to simplify the computation of the modulation func-tions terms such as

‖F−11,i (τs)L−1

i+1,N (s)ω‖ ≤ cωis + λLi

‖ω‖ , (i = 0, . . . , N − 1) . (87)

The above bound was obtained by application of Corollary 2 (see Appendix A).An important point is that, combined with (86), it does not require the regres-sor vector to be computed. This certainly eliminates a lot of computationalburden.

Relay UV-MRAC

Significant simplification is possible if local stability is sufficient for a partic-ular application. It consists in using constant control amplitudes -i ≡ ci (i =0, . . . , N). Then, the filters applied in the computation of the modulationfunctions are not needed. Tracking with disturbance rejection can be ob-tained with appropriate amplitudes ci. The resulting controller is denotedRelay UV-MRAC because in the SISO case the control signals are generatedby simple relays Ui = cisgn(εi) [19].

8 Simulation Results

A simulation example is presented in order to evaluate the performance ofthe UV-MRAC. No nominal parameter matrix is applied (θnom = 0) to allowthe design of the controller to be carried out with minimum knowledge aboutthe plant.

302 L. Hsu et al.

An articulated suspension system is described in Fig. 4. The objective isto make the load position (y2) and orientation angle (φ) track the referencemodel output through the control of two linear displacement actuators. Thecontrol signals are the actuator forces u1 and u2. The plant is linearized inthe neighborhood of φ = 0 rad, resulting in the input-output representation

y = diag

1s2

,1s2

Kp[u + d] , (88)

Kp =

[l1/J −l2/J

1/m 1/m

], d = K−1

p

[0g

], u = u− dnom , (89)

where the plant output vector is y = [φ y2]T and the control vector is u =[u1 u2]T . The term dnom = [50 50]T N is used for gravity compensation, thusallowing the reduction of the amplitude of the controller signal u. We setd0(t) ≡ 250 and d1(t) ≡ 50 in the modulation functions to account for theresidual disturbance term (d− dnom).

Fig. 4. Diagram of the suspensionsystem

Fig. 5. Output error signals for thesuspension system

Fig. 6. Load orientation angle andmodel output (φ, φM )

Fig. 7. Load position and model out-put (y2, yM2)

The parameters of the plant are: load mass m = 10 kg, load momentof inertia J = 1 kgm2, gravity acceleration g = 9.81m/s2, platform length


l = 4m and load position l2 = 3m. Such parameters were not used in theUV-MRAC design. The platform mass, the actuator mass and the frictionare neglected.

The reference model is WM (s) = [(s+1)(s+5)]−1I. The reference signalsr1 and r2 are a sine wave of amplitude 0.1 and frequency 1 rad/s and a squarewave of amplitude 0.1 and frequency 2 rad/s, respectively. The controllerparameters are: cω0 = cU0 = 1, cω1 = 0.2, δ0 = δ1 = 0.1, L(s) = Λ(s) = s+5,Knom = 0.1I and Sp = I.

The convergence of the error signals observed in Fig. 5 is exponentiallyfast. However, quite large peaks, of amplitudes 0.25 rad (≈ 14) in e1 and0.1m in e2, are present in the beginning of the convergence phase due tounfavorable initial conditions. This justifies the practice of initializing thereference model such that the initial error is small.

The Hurwitz condition required to apply the UV-MRAC is satisfied forSp = I if and only if l1 > −J/m, which is always true. However, if thecontrol law requires that KpSp to be positive definite, such as in [37,7], thenthe necessary and sufficient conditions are l1 > −J/m and

l22 +2Jm

l2 +J2

m2− 4Jl

m< 0 . (90)

In this numerical example the load position should be kept within l2 <1.164 to allow the application of [37,7], otherwise, an appropriate Sp ma-trix should be chosen to make KpSp positive definite. It should be stressedthat the positive definiteness of KpSp implies that (−KpSp) be Hurwitz, butthe converse is not true, thus the UV-MRAC can control plants with HFGbelonging to a broader class of matrices.

The averaging filters time constant (τ = 0.003 s) was chosen small enoughto keep the closed loop system stable and the output error small.

The constants in the modulation functions (cω0, cω1 and cU0) are such thatthe amplitude of the control signal u is as small as possible, but the controllerstability and performance are robust to the plant parameter uncertainties.

9 Conclusion

A model-reference sliding mode control strategy for uncertain linear MIMOsystems by output-feedback was presented. Although the derivation of thenew controller, called UV-MRAC, follows in essence its SISO counterpart,the development of some new results for unit vector control systems wasnecessary. A notable stability condition, which already appeared in differentcontexts of Control Theory, was that the plant high frequency gain (HFG)matrix, possibly compensated by some static matrix gain, should be Hurwitz.This condition is more stringent than the simple knowledge of the sign of theHFG in the SISO case. Yet, for the MIMO case it does not seem to be overly

304 L. Hsu et al.

restrictive since it is known to be a necessary and sufficient condition for theexistence of sliding modes with unit vectors as switching control laws.

The new controller UV-MRAC utilizes averaging filters with sufficientlysmall time constant τ . This time constant is similar to the small parameterε that characterizes the high gain observers (HGO) of output-feedback slid-ing mode controllers. In the latter, peaking of control signals and observervariables may arise as ε → 0. In both types of controllers, tracking errorstend to zero as the small parameters tend to zero. However, in contrast tothe HGO case, an important feature of the UV-MRAC is that it possessesglobal exponential properties uniformly with respect to τ ∈ (0, τ∗] for somesmall enough τ∗. This implies that the UV-MRAC preserves global stabilityand is free of peaking as τ → 0.

10 Acknowledgment

This work was partially supported by CNPq, FAPERJ, PRONEX/FINEPand FUJB (Brazil).

Appendix

A Complementary Lemmas

Lemma 3. Let r(t) be an absolutely continuous scalar function. Suppose r(t)is nonnegative and while r > 0 it satisfies r ≤ −δ−γr+Re−λt, where δ, γ, λ,Rare nonnegative constants. Then, one can conclude that: (a) r(t) is boundedby

r(t) ≤ [r(0) + cR]e−min(λ,γ)t (∀t ≥ 0), (91)

where c is an appropriate positive constant; (b) if δ > 0 then there existsts < +∞ such that r(t) ≡ 0 for all t ≥ ts.

Proof. According to the Comparison Theorem [12], an upper bound r(t) ofr(t), is given by the solution of:

˙r = −γr + Re−λt , r(0) = r(0) .

Hence, r(t) ≤ [r(0)+cR]e−min(γ,λ)t, which proves the first part of the Lemma.Now, if δ > 0, the comparison equation is

˙r = −δ − γr + Re−λt, r(0) = r(0) , (92)

which implies that while r(t) > 0 one has r(t) ≤ [r(0)+δ/γ+cR]e−min(γ,λ)t−δ/γ. Defining ts = [min(λ, γ)]−1lnγ(r(0) + δ/γ + cR)/δ < +∞, one notesthat the right hand side of the latter inequality is negative ∀t > ts. Thus,since r(t) continuous and r(t) ≥ 0 one concludes that r(t) becomes identicallyzero for t ≥ ts.


Lemma 4. Consider the system

z = W (s)d , z ∈ Rm, d ∈ R

p , (93)

where W (s) is a stable and strictly proper m×p transfer matrix. Let γ0 be thestability margin of W (s), i.e., 0<γ0<minj |Re(pj)| , where pj are the poles ofW (s). Let d(t) be an instantaneous upper bound of d(t), i.e., ‖d(t)‖ ≤ d(t) ∀t.Then, there exists a positive constant c1 such that the impulse response w(t)satisfies ‖w(t)‖ ≤ c1γ0e

−γ0t and the following inequalities hold

‖w(t) ∗ d(t)‖ ≤ c1γ0e−γ0t ∗ d(t) = c1

γ0

s + γ0∗ d(t) , (94)

‖z(t) − z0(t)‖ ≤ c1‖df (t) − d0f (t)‖ , df = (

γ0

s + γ0)d , (95)

‖z(t)‖ ≤ c1df (t) + exp , (96)

where z0, d0f and “exp” depend on the initial conditions and decay exponen-

tially to zero with rate γ0.

Proof. The proof follows from a direct extension of the scalar case in [25].

Corollary 2 (Lemma 4). Consider

z = GF (τs)GL(s)d = GF (τs)1

s + αGL(s)d , (97)

where GF , GL are rational, stable, strictly proper, GL has positive impulseresponse, α > 0 is the stability margin of GL(s). If τ ∈ [0, τ ] and τ is suffi-ciently small, there exists k > 0 such that (95) and (96) hold with

df (t) = k1

s + αd(t) . (98)

Proof. The proof follows from a direct extension of the scalar case in [23].

B Proof of Lemmas and Theorems

B.1 Proof of Proposition 1

Since −K is Hurwitz, there exists P = PT > 0 and Q = QT > 0 such thatKTP +PK = Q. Thus, consider the quadratic form V (ε) = εTPε which hastime derivative bounded by

V ≤ −-λmin(Q)‖ε‖ + λmax(ATP+PA)‖ε‖2+ 2σmax(PK)‖ε‖ (‖d‖+‖π‖)Now, choosing - as in (25) with

cε ≥ maxλmax(ATP+PA)

λmin(Q)+ δ , 0

, cd ≥ 2

σmax(PK)λmin(Q)

− 1 , (99)

306 L. Hsu et al.

where the constant δ > 0 provides some desired stability margin, we obtain

V ≤ −λmin(Q)[δ + δ‖ε‖ − (1 + cd)Re−λt

] ‖ε‖ . (100)

Now, from the Rayleigh-Ritz inequality λmin(P ) ‖ε‖2 ≤ V (ε) ≤ λmax(P ) ‖ε‖2,and denoting cQ1 = λmin(Q)/

√λmax(P ) (> 0), cQ2 = λmin(Q)/λmax(P ) (>

0) and cD = (1 + cd)λmin(Q)/√

λmin(P ) (> 0), inequality (100) can berewritten as

V ≤ −δ cQ1

√V − δ cQ2V + cDRe−λt

√V . (101)

Then, defining r :=√V , one obtains

2r ≤ −δ cQ1 − δ cQ2 r + cD Re−λt . (102)

Thus from Lemma 3 (see Appendix A), we can conclude that r(t) ≤ [r(0) +cR]e−λ1t , where c > 0 is an appropriate constant and λ1 = min

(λ, δ cQ2/2

).

Applying the Rayleigh-Ritz inequality, we finally obtain inequality (26). Ifδ > 0 in (25), from Lemma 3, one can further conclude that there existst1 < +∞ such that r(t) ≡ 0, ∀t > t1, hence, the sliding mode at ε = 0 startsin some finite time ts, 0 ≤ ts ≤ t1.

B.2 Proof of Lemma 1

Consider a stabilizable and detectable realization of (27)

x = Ax + B(u + d + π) , ε = Cx . (103)

The high frequency gain is K = CB. System (103) can be transformed tothe regular form [39]

x1 = A11x1 + A12ε , (104)ε = A21x1 + A22ε + K(u + d + π) . (105)

The state vector of this realization is xTe = [xT1 εT ]. The zero dynamics isgiven by x1 = A11x1. Since M(s) is minimum phase, A11 is Hurwitz. Let γ0

be the stability margin of A11. Then, according to Lemma 4 applied to (104)written as x1 = (sI −A11)−1A12ε, one has the bound

‖x1(t)‖ ≤ c1[ |εf (t)| + |ε0f (t)| ] + ‖x0

1(t)‖ , εf =γ0

s + γ0∗ ‖ε‖ , (106)

where ε0f and x0

1 are exponentially decaying terms due to initial conditions.On the other hand, since −K is Hurwitz, and thus nonsingular, equation(105) can be rewritten as

ε = A22ε + K[u1 + d + K−1A21(sI−A11)−1A12 ∗ ε + π1

], (107)


where u1 = −-1ε

‖ε‖ and -1 = (1 + cd)‖d(t)‖ + cε‖ε‖ + cfγ0

s+γ0∗ ‖ε‖. The

term π1 takes into account the terms π, x01 and ε0

f and is bounded by ‖π1‖ ≤[c1‖xea(0)‖ + R]e−λ1t, where xea = [xT1 εT εf ]T is the complete state vectorof the closed loop system and λ1 = min(γ0, λ). Thus, from Proposition 1,inequality (30) is satisfied for ‖ε‖, provided that cd, cε, cf ≥ 0 are appropri-ately chosen. Since, ‖x1‖ and ‖εf‖ satisfy similar inequality, then ‖xea(t)‖also satisfies (30).

Also from Proposition 1, one can further conclude that ε becomes identi-cally zero after some finite time ts, provided that δ > 0 in (28).

B.3 Proof of Corollary 1

For simplicity, we will consider a controllable realization. In this case, if thereare unobservable states, the element A21 of the regular form (104)–(105) isidentically zero, i.e., A21 = 0. Then, the result follows directly from the proofof Lemma 1. In the case of a nonminimal realization which is noncontrolableand/or nonobservable, the proof follows in a similar way, using the KalmanDecomposition.

Now, if αi = α, (∀i), then A22 = −αI satisfies the Lyapunov equationPA22+AT

22P = −Q2, Q2 > 0 for any P = PT > 0. Since the constant cε, inLemma 1, comes from inequality (99) in Proposition 1, one can chose cε = 0.

B.4 Proof of Lemma 2

The time derivative of the quadratic form V (t)= εTP ε (P =PT >0) is givenby

V = −2αεTP ε + [εTPKu + uTKTP ε] + 2εTPKd .

Then, with u = −- ε‖ε‖ , we have that

V = −2αV − -εTPKε + εTKTP ε

‖ε‖ + 2εTPKd .

Now, define the auxiliary functions

Vs = −2αVs , V = Vs + γ ,

where γ is an absolutely continuous function satisfying γ ≥ −2αγ and γ ≥c2‖β+π‖2 for some positive constant c, yet to be defined. Since ˙V = −2αVs+γ = −2α(V − γ) + γ, we can write the following comparison equation

d

dt[V − V ] = −2α[V − V ] + γ + 2αγ + 2-

εTPKε

‖ε‖ − 2εTPKd . (108)

308 L. Hsu et al.

Considering γ given by γ(t) = c2[R e−min(α,λ)t + ‖βt‖∞]2, it is easy to verifythat Vs(0) > 0 implies that

Vs(t) = e−2αtVs(0) > 0 , (109)

V (t) > c2[R e−min(α,λ)t + ‖βt‖∞]2 , ∀t . (110)

Then, if V (t0) = V (t0) for any t0 ≥ 0, the inequality (110) implies thatV (t0) > c2[R e−min(α,λ)t0 + ‖βt0‖∞]2 (∀t0 ≥ 0). From the Rayleigh-Ritzinequality, λmin(P )‖ε‖2 ≤ V ≤ λmax(P )‖ε‖2, one concludes that

‖ε(t0)‖ ≥ c√λmax(P )

[R e−min(α,λ)t0 + ‖βt0‖∞] (111)

and also, for t = t0,

d

dt[V − V ] ≥ 2-

εTPKε

‖ε‖ − 2εTPKd . (112)

Since KTP + PK = Q > 0, we can rewrite (112) as

d

dt[V − V ] ≥ -

εTQε− 2εTKTP (β + π)‖ε‖ − 2εTPKd

and consequently, denoting cd = 2σmax(PK)/λmin(Q) − 1 (≥ 0),

d

dt[V − V ] ≥ λmin(Q)

[-‖ε‖ − (cd + 1)‖β + π‖

‖ε‖ − (cd + 1)‖d‖]‖ε‖

≥ λmin(Q)[-

(1 − (cd + 2)

‖β + π‖‖ε‖

)− (cd + 1)‖d‖

]‖ε‖ .

Then, choosing c ≥ (1 + k)(cd + 2)√

λmax(P ) in (111) for some k > 0, onehas

‖β(t0) + π(t0)‖‖ε(t0)‖ ≤ 1

(1 + k)(cd + 2). (113)

Therefore, we have for t = t0,

d

dt[V − V ] ≥ λmin(Q)

[-

k

1 + k− (cd + 1)‖d‖

]‖ε‖ .

Now, if we choose -(t) such that -(t) ≥ (cd + 1)‖d(t)‖ , ∀t , where cd =(1 + k−1)(cd + 1) − 1 > 0, we have that, for t = t0, d

dt [V − V ] ≥ 0. Thususing the Comparison Theorem in [12, Theorem 7], we conclude that V (t0) ≤˙V (t0) implies V (t) ≤ V (t), ∀t ≥ t0. Then, from the Rayleigh-Ritz inequalityand since V (t) = e−2αtVs(0) + γ(t), we obtain inequality (34), with c1 =√

λmax(P )/λmin(P ) and, without loss of generality, t0 =0.


B.5 Proof of Theorem 4

By Corollary 1, ε0 (63), as well as the state Xε of (63), converge to zero, atleast exponentially, if the signal -0 satisfies (72). Note that the transient partof L−1 is represented by π(t) which is thus bounded by Π0. Consequently,one concludes ‖ε0(t)‖ ≤ Π0.

From (70) one can write π0i = Hi(s)ε0 = hi(t) ∗ ε0(t) + π00i(t). Since

‖π00i‖ ≤ Π0 and ‖hi(t) ∗ ε0(t)‖ ≤ Π0, then ‖π0i‖ ≤ Π0. Now, for N > 1 and

i = 1, since πe1 ≡ 0, equation (65) results in ‖ε1(t)‖ ≤ Π0, from Lemma 2with β(t) ≡ 0. With similar argument, we recursively conclude from (65),(69) and Lemma 2 that ‖πei(t)‖ ≤ Π0 and ‖εi(t)‖ ≤ Π0 (i = 2, . . . , N − 1).

Now consider εN (see (66)). Note that, from assumption (A7) (see Sect. 5),Mθ and Mred can be chosen so that ‖UN (t)t‖∞ ≤ C(t). Since πe,N−1 andεN−1 are bounded by Π0, so is πeN , i.e., ‖πeN (t)‖ ≤ Π0. From (68) one has

‖(βuN − β0uN )t‖∞ ≤

∥∥∥(F1,N (τs) − I)F−11,N (τs)L−1

N (s)∥∥∥︸︷︷︸

O(τ)

C(t)

= τKβNC(t) , (114)

where β0uN (t) is bounded by Π0. Then, applying Lemma 2 to equation (66),

which can be rewritten as εN = −L−1N (Knom)−1K[UN + F−1

1,NUd] − (I −(Knom)−1K)[βuN − β0

uN ]− [(I − (Knom)−1K)β0uN + πeN + π0N ], one readily

concludes (74). Since ‖(βuN − β0uN )t‖∞ ≥ ‖βuN−β0

uN‖ ≥ ‖βuN‖ −Π0, then(76) follows from (114).

B.6 Proof of Theorem 5

It is convenient to rewrite (66) as

εN = L−1N [−UN − U ] + βuN − πeN − π0N , (115)

where βuN = (F1,N−I)F−11,NL−1

N U can be bounded by C(t) (see (77)) similarlyas βuN in (114), i.e., ‖(βuN − β0

uN )t‖∞ ≤ τKβNC(t). Remembering thatu = unom − SpUN , we note that L−1

N in (115) operates on the same signalUN as the one in (50). From (50) and (66), the model following error can berewritten as: Xe = AcXe + BcK

nom[˙eN + αN eN

], where eN :=εN−(βuN −

πeN − π0N ). To eliminate the derivative term ˙eN , a variable transformationXe := Xe −BcK

nomeN is performed yielding˙Xe = AcXe + (Ac + αNI)BcK

nomeN . (116)

The bound (74) (Theorem 4) and the exponential stability of Ac imply thatXe(t) is bounded by ‖Xe(t)‖ ≤ τKC(t) + Π. Moreover, as explained below,

‖Xe(t)‖ ≤ τKeC(t) + Π , and ‖e(t)‖ ≤ τK0C(t) + Π , (117)‖ωt‖∞ ≤ τK1C(t) + K2‖z(0)‖ + Km , (118)

C(t) ≤ K ′red + K4‖z(0)‖

1 − τK3. (119)

310 L. Hsu et al.

Indeed, inequalities (117) follow from Xe=Xe−BcKnomeN . From the relation

(47), it follows that ‖ω‖≤KM+KΩ‖Xe‖ and, then from (117) we obtain (118),where K2‖z(0)‖ comes from the initial value of the term Π appearing in thebound (117). Now, from (77) and (118), C(t)≤ τK3C(t)+K4‖z(0)‖+K ′

red,whereby, after a simple algebraic manipulation one obtains (119), which isvalid for τ <K−1

3 . Now, as explained below, we can also write

‖z0(t)‖ ≤ Kze−azt‖z0(0)‖ , (120)

‖ze(t)‖ ≤ τK5

(‖ze(0)‖ + ‖z0(0)‖)+ O(τ) + Π . (121)

Indeed, the variables Xε, εi=0,...,N−1, x0FL are bounded by Π0 in Theorem 4.

Therefore, with the partition zT = [(z0)T , zTe ], where zTe := [εTN ,XTe ] one gets

(120), where only the initial condition on z0 appears. Now, from (74), (117)and (119) follows (121), where O(τ) is independent of the initial conditions.Noting that the initial time is irrelevant in deriving the above expressions, wecan write, for arbitrary t ≥ tk ≥ 0 (k = 0, 1, . . . ) and some T1 = tk+1−tk > 0

‖ze(t)‖≤[τK5+K6e

−a(t−tk)] [‖ze(tk)‖+‖z0(tk)‖

]+O(τ) , (122)

‖z0(t)‖≤Kze−az(t−tk)‖z0(tk)‖ , (123)

‖ze(tk+1)‖ ≤ λ(‖ze(tk)‖ + ‖z0(tk)‖

)+ O(τ) , (124)

‖z0(tk+1)‖ ≤ λ‖z0(tk)‖ . (125)

Equations (124) and (125) are obtained from (122) and (123) as follows: forτ <K−1

5 , there exists T1>0 such that λ=max(τK5 +K6e−aT1 ,Kze

−azT1) <1. Then, the simple linear recursive inequalities (124) and (125) hold andeasily lead to the conclusion that, for τ small enough, the error system isglobally exponentially stable with respect to a residual set of order τ .

References

1. Ambrosino, G., G. Celentano, and F. Garofalo (1984). Variable structureMRAC systems. Int. J. Contr. 39, 1339–1349

2. Baida, S. V. (1993) Unit sliding mode control in continuous- and discrete-timesystems. Int. J. Contr. 57, 1125–1132

3. Bartolini, G. and T. Zolezzi (1988) The V.S. approach to the model referencecontrol of nonminimal phase linear plants. IEEE Trans. Aut. Contr. 33, 859–863

4. Bondarev, A. G., S. A. Bondareva, N. E. Kostyleva, and V. I. Utkin (1985)Sliding modes in systems with asymptotic state observers. Autom. RemoteControl Pt. 1. 46, 679–684

5. Byrnes, C. I. and J. C. Willems (1984) Adaptive stabilization of multivariablelinear systems. In Proc. IEEE Conf. on Decision and Control, Las Vegas, NV,1574–1577

6. Chien, C.-J. and L.-C. Fu (1992) A new robust model reference control withimproved performance for a class of multivariable unknown plants. Int. J. Adap-tive Contr. Signal Process. 6, 69–93


7. Chien, C.-J., K.-C. Sun, A.-C. Wu, and L.-C. Fu (1996) A robust MRAC usingvariable structure design for multivariable plants. Automatica 32, 833–848

8. DeCarlo, R. A., S. H. Zak, and G. P. Matthews (1988) Variable structure controlof nonlinear multivariable systems: a tutorial. Proceedings of the IEEE 76,212–232

9. Elliott, H. and W. A. Wolovich (1982) A parameter adaptive control structurefor linear multivariable systems. IEEE Trans. Aut. Contr. 27, 340–352

10. Emelyanov, S. V., Korovin, Nersisian, and Nisezon (1992) Discontinuous outputfeedback stabilizing an uncertain MIMO plant. Int. J. Contr. 55, 83–109

11. Esfandiari, F. and H. K. Khalil (1992) Output feedback stabilization of fullylinearizable systems. Int. J. Contr. 56, 1007–1037

12. Filippov, A. F. (1964) Differential equations with discontinuous right-hand side.American Math. Soc. Translations 42, 199–231

13. Filippov, A. F. (1988) Differential Equations with Discontinuous RighthandSides. Kluwer Academic Publishers

14. Gibbens, P. W., C. A. Schwartz, and M. Fu (1993) Achieving diagonal in-teractor matrix for multivariable linear systems with uncertain parameters.Automatica 29, 1547–1550

15. Gutman, S. (1979) Uncertain dynamical systems — a Lyapunov min-max ap-proach. IEEE Trans. Aut. Contr. 24, 437–443

16. Gutman, S. and G. Leitmann (1975) Stabilizing control for linear systems withbounded parameters and input uncertainty. In Proc. 7th IFIP Conf. on Opti-mization Techniques, Nice, France, 8–14

17. Hsu, L. (1990) Variable structure model reference adaptive control using onlyI/O measurement: General case. IEEE Trans. Aut. Contr. 35, 1238–1243

18. Hsu, L. (1997) Smooth sliding control of uncertain systems based on a predic-tion error. Int. J. on Robust and Nonlinear Control 7, 353–372

19. Hsu, L., A. D. Araujo, and R. R. Costa (1994) Analysis and design of I/Obased variable structure adaptive control. IEEE Trans. Aut. Contr. 39, 4–21

20. Hsu, L. and R. R. Costa (1989) Variable structure model reference adaptivecontrol using only input and output measurement: Part I. Int. J. Contr. 49,399–416

21. Hsu, L., R. R. Costa, A. K. Imai, and P. Kokotovic (2001) Lyapunov-basedadaptive control of MIMO systems. In Proc. American Contr. Conf., Arlington,VA, 4808–4813

22. Hsu, L., E. Kaszkurewicz, and A. Bhaya (2000) Matrix-theoretic conditions forthe realizability of sliding manifolds. Systems & Contr. Letters 40, 145–152

23. Hsu, L., F. Lizarralde, and A. D. Araujo (1997) New results on output-feedbackvariable structure adaptive control: design and stability analysis. IEEE Trans.Aut. Contr. 42, 386–393

24. Ilchmann, A. and D. H. Owens (1990) Adaptive stabilization with exponentialdecay. Systems & Contr. Letters 14, 437–443

25. Ioannou, P. and K. Tsakalis (1986) A robust direct adaptive controller. IEEETrans. Aut. Contr. 31, 1033–1043

26. Ioannou, P. A. and J. Sun (1996) Robust Adaptive Control. Prentice-Hall27. Kailath, T. (1980) Linear Systems. Prentice Hall28. Lu, X.-Y. and S. K. Spurgeon (1999) Output feedback stabilization of MIMO

non-linear systems via dynamic sliding mode. Int. J. of Nonlinear and RobustControl 9, 275–305

312 L. Hsu et al.

29. Narendra, K. and A. Annaswamy (1989) Stable Adaptive Systems. PrenticeHall

30. Oh, S. and H. K. Khalil (1995) Output feedback stabilization using variablestructure control. Int. J. Contr. 62, 831–848

31. Sastry, S. S. and M. Bodson (1989) Adaptive Control: Stability, Convergenceand Robustness. Prentice-Hall

32. Schwartz, C. A. and A. Yan (1995) Comments on ‘achieving diagonal interactormatrix for multivariable linear systems with uncertain parameters’. Automat-ica 31, 161–164

33. Shyu, K.-K., Y.-W. Tsai, Y. Yu, and K.-C. Chang (2000) Dynamic outputfeedback sliding mode design for a class of linear unmatched uncertain systems.Int. J. Contr. 73, 1463–1474

34. Slotine, J. J. E., J. K. Hedrick, and E. A. Misawa (1987) On sliding observers fornonlinear systems. ASME J. Dynamic Systems Measurement and Control 109,245–252

35. Spurgeon, S. K., C. Edwards, and N. P. Foster (1996) Robust model referencecontrol using a sliding mode controller/observer scheme with application to ahelicopter problem. In Proc. IEEE Workshop on Variable Struct. Sys., 36–41

36. Tao, G. and P. A. Ioannou (1988) Robust model reference adaptive control formultivariable plants. Int. J. Adaptive Contr. Signal Process. 2, 217–248

37. Tao, G. and P. A. Ioannou (1989) A MRAC for multivariable plants with zeroresidual tracking error. In Proc. IEEE Conf. on Decision and Control, Tampa,1597–1600

38. Utkin, V. I. (1983) Variable structure systems: Present and future. Automationand Remote Control 44, 1105–1120

39. Utkin, V. I. (1992) Sliding Modes in Control Optimization. Springer-Verlag40. Walcott, B. L. and S. Zak (1988) Combined observer-controller synthesis for

uncertain dynamical systems with applications. IEEE Trans. Syst. Man andCyber. 18, 88–104

41. Wolovich, W. A. and P. L. Falb (1976) Invariants and canonical forms underdynamic compensation. SIAM J. Contr. Optim. 14, 996–1008

42. Young, K.-D. D. (1977) Asymptotic stability of model reference systems withvariable structure. IEEE Trans. Aut. Contr. 22, 279–282

43. Young, K. K. D., V. Kokotovic, and V. Utkin (1977) A singular perturbationsanalysis of high-gain feedback systems. IEEE Trans. Aut. Contr. 22, 931–939

44. Zak, S. H. and S. Hui (1993) Output feedback variable structure controllers andstate estimators for uncertain/nonlinear dynamic systems. IEE Proc.-D. 140,41–50

45. Zinober, A. S. I., O. M. E. El-Ghezawi, and S. A. Billings (1982) Multivari-able variable structure adaptive model–following control systems. IEE Proc.-D. 129, 6–12


Sliding Modes, Differential Flatness andIntegral Reconstructors

Hebertt Sira-Ramırez1 and Victor M. Hernandez2

1 CINVESTAV-IPNDepartamento de Ingenierıa Electrica, Seccion de MecatronicaAvenida IPN, No. 2508Colonia San Pedro Zacatenco AP 1474007300 Mexico, D.F., Mexico

2 Universidad Autonoma de QueretaroFacultad de IngenierıaCentro Universitario, Cerro las Campanas76010 Queretaro, Mexico

Abstract. The relevance of the differential flatness property in the context of slid-ing mode controller design is explored for the case of linear and nonlinear SISOand MIMO systems. We also explore, for linear systems, the possibilities of ex-tending the idea of Generalized PI control, based on state reconstructors, to theproblem of sliding surface asymptotic synthesis requiring no state measurements,or asymptotic observer design. An experimental example is presented, dealing withthe sliding mode control of an electro-mechanical system, where there is no needfor mechanical sensors.

1 Introduction

In this chapter we examine the relevance of the differential flatness propertyin the design of sliding mode controllers for nonlinear Single-Input Single-Output (SISO) and Multiple-Input Multiple-Output (MIMO) systems. It isshown that flatness considerably simplifies the sliding mode controller designprocess by reducing the problem to that of controlling a linearized systemin Brunovsky’s canonical form in the case of SISO systems or, in the MIMOcase, to that of controlling a suitable set of independent integrator chainsplaced also in Brunovsky’s canonical form.

A new development, regarding the possibilities of sliding mode controlsynthesis without state measurements is envisioned from the perspective of therecently introduced Generalized PI control or, more properly called, “IntegralReconstruction” schemes for state feedback policies (see Fliess et al [2]-[4]).

Section 2 introduces in a tutorial manner the concept of differential flat-ness for nonlinear systems. Some illustrative examples are presented regard-ing the uncovering of this important, though non-generic, property. Section3 presents the sliding mode control of flat systems from the perspective ofGeneralized PI control. Here, we consider the use of integral reconstructorsin the stabilization and trajectory tracking, without state measurements, of


316 H. Sira-Ramırez and V. Hernandez

linearized flat systems. Section 4 contains the results of an actual experimen-tal application of GPI sliding mode control to a linear electro-mechanicalsystem. The conclusions and suggestions for further work are collected at theend of the chapter.

2 Sliding Mode Control of Differentially Flat Systems

2.1 Differential Flatness

Differential flatness was introduced by Prof. M. Fliess and his co-workersin a series of interesting articles nearly 10 years ago (see Fliess et al , [2],and the references therein). Important contributions to the subject have alsobeen given by Pomet [9], Rathinham [10], Sluis [13] and Nieuwstadt [8]. Theconcept of flatness arose from the early work of E. Cartan [1] in connectionwith studies of underdetermined systems of differential equations.

Roughly speaking, an n-dimensional system, of the form x = f(x, u), y =h(x), with multiple outputs denoted by the vector y which is provided withm independent inputs u, entering regularly into the system equations, is dif-ferentially flat if we can find m artificial outputs F of the form:

F = ψ(x, u, u, ..., u(α))

such that all variables in the system, i.e., states, original outputs, and all con-trol inputs, can be written in terms of the vector F and its time derivatives,without integrating any differential equations. In other words a differentialparameterization is possible for all system variables in terms of F alone, i.e.

x = Φ(F, F , ...., F (β)) y = Ψ(F, F , ..., F (β))u = Θ(F, F , ..., F (β+1))

where β is to be understood as a vector of integers containing in each entrythe order of derivation of the corresponding component of the vector F , i.e.

F (β) = (F (β1)1 , ..., F (βm)

m )

It is clear that the flatness property trivializes the feedback linearizationproblem. For instance, for the special case in which the function Θ is locallyregular with respect to F (β+1), the local input coordinate transformation

u = Θ(F, F , ..., F (β), v) (1)

takes the system into a set of decoupled systems in Brunovsky’s canonicalform:

F(βi+1)i = ϑi, i = 1, ...,m

When the function Θ is not regular in F (β+1) then, generally speaking,an obstruction exists for static feedback linearization and further derivations

Sliding Modes, Differential Flatness, and Integral Reconstructors 317

must be taken in some of the control input expressions, i.e. in some of thecomponents of the mapping Θ, so as to achieve the required regularity andlocal invertibility. This procedure, modulo possible singularities, leads to dy-namic feedback linearization achievable in an endogenous manner.

2.2 Sliding mode control based on flatness

It is clear that, in the regular case, a sliding mode controller for the nonlinearsystem can also be prescribed in terms of the flat outputs by defining a set ofm suitable sliding surfaces coordinate functions, denoted by σi, i = 1, ...,m,specified on the basis of the state variables characterizing each one of theindependent Brunovsky chains of integrators, i.e. we may adopt as slidingsurface coordinate functions,

σi =βi+1∑j=1

αji

(F

(j−1)i − [Fi

∗(t)](j−1))

for suitable Hurwitz coefficients αji, j = 1, ..., βi+1, with, αβi+1,i = 1, for alli, and where the signals: Fi

∗(t), i = 1, ...,m, represent the desired trajectoriesfor the flat outputs.

The sliding surface is defined as the non-empty intersection of the zerolevel sets of all the sliding surface coordinate functions.

The time derivative of the i-th sliding surface coordinate function is ob-tained as

σi =βi+1∑j=1

αji

(F

(j)i − [Fi

∗(t)](j))

=βi∑

j=1

αji

(F

(j)i − [Fi

∗(t)](j))+ ϑi − [Fi

∗(t)](βi+1)

The discontinuous feedback control actions for the auxiliary control vari-ables, ϑi, i = 1, ...,m, is prescribed to be,

ϑi = −Wisign σi, i = 1, ...,m

with, Wi = δi+ηi, i = 1, 2, ...m where ηi > 0 and, δi, chosen so as to satisfy,

δi > | − [Fi∗(t)](βi+1) +

βi∑j=1

αji

(F

(j)i − [Fi

∗(t)](j))

| ∀ t, i = 1, 2, ...,m

These choices result in the following inequalities for each sliding surface co-ordinate function, σi,

12

d

dt

(σ2

i

) ≤ −ηi| σi |, i = 1, 2, ...,m


thus yielding a local sliding motion on the intersection manifold:

S =m⋂

i=1

x : σi = 0

The actual feedback control inputs u are obtained by reversing the regularinput coordinate transformation (1).

Example 1. Extended two wheeled non-holonomic carConsider the following model of a non-holonomic two wheeled car

x = v cos θ, y = v sin θ, θ = u2, v = u1

The system is differentially flat, with flat outputs given by the wheel’s com-mon axis midpoint coordinates (F1, F2) = (x, y). Indeed, all system variablesare differentially parameterizable in terms of F1, F2 and a finite number oftheir time derivatives

x = F1, y = F2, θ = arctan

(F2

F1

), v =

√F 2

1 + F 22

u1 =F1F1 + F2F2√

F 21 + F 2

2

, u2 =F1F2 − F2F1

F 21 + F 2

2

The parameterization of the control inputs is locally regular with respect tothe second order derivatives of the flat outputs F1, F2.

[u1

u2

]=

F1√F 2

1 + F 22

F2√F 2

1 + F 22

− F2

F 21 + F 2

2

F1

F 21 + F 2

2

[F1

F2

], det

(∂u

∂F

)=

1√F 2

1 + F 22

The locally non-singular input coordinate transformation:

[u1

u2

]=

F1√F 2

1 + F 22

F2√F 2

1 + F 22

− F2

F 21 + F 2

2

F1

F 21 + F 2

2

[ϑ1

ϑ2

]

transforms the system into the following set of two decoupled integratorchains in Brunovsky form,

F1 = ϑ1, F2 = ϑ2

Suppose it is desired to follow a smooth desired trajectory (x∗(t), y∗(t),for the position variables (x, y). The sliding surface coordinate functions,

σ1 = F1 − x∗(t) + λ1(F1 − x∗(t)), λ1 > 0σ2 = F2 − y∗(t) + λ2(F2 − y∗(t)), λ2 > 0


impose independent exponentially asymptotically stable motions towards thedesired trajectories upon permanently enforcing the ideal sliding conditions:σ1 = 0, σ2 = 0, by means of the auxiliary control inputs ϑ1 and ϑ2. The timederivatives of the sliding surface coordinates σ1 and σ2 are given by:

σ1 = ϑ1 − x∗(t) + λ1

(F1 − x∗(t)

)

σ2 = ϑ2 − y∗(t) + λ2

(F2 − y∗(t)

)

The choicesϑ1 = −W1 sign σ1, ϑ2 = −W2 sign σ2

with Wi = δi + ηi, i = 1, 2 and ηi > 0, i = 1, 2, where δ1, δ2 satisfy:

δ1 > | − x∗(t) + λ1

(F1 − x∗(t)

)| > 0 ∀ t

δ2 > | − y∗(t) + λ2

(F2 − y∗(t)

)| > 0 ∀ t

yield, ddt

(σ2

i

) ≤ −ηi|σi|, i = 1, 2. Thus, the prescribed discontinuous feed-back control laws locally create sliding motions on the intersection of themanifolds, σ1 = 0 and σ2 = 0.

However, due to the fact that the actual control, u2, is a velocity, whichevidently cannot be dicontinuous, we use a ”smoothed” sign function for thespecification of the auxiliary control input, ϑ2, in the form: ϑ2 = σ2/(ε2+|σ2|)with ε2 being a small strictly positive number. Similarly, for the choice of thecontrol input ϑ1, we take ϑ1 = σ1/(ε1 + |σ1|). Note that this input is, in fact,an acceleration and can, in principle, be chosen as a discontinuous function.Evidently these “high gain ” choices compromise the steady state values ofthe tracking errors in the presence of unmodeled perturbations.

Figure 1 shows the performance of the high gain multivariable controllersolving the task of having the two wheeled car follow a three petals rosefigure in the plane. Such a graph is defined in the plane, (x, y), by the polarcoordinates equation, ρ(t) = R cos(3ψ(t)) with (ρ, ψ) being, respectively, theradius vector and the angular position of the radius vector, with respect tothe x axis, characterizing a point located on the rose graph. We adopt thefollowing time parameterization for the angular position coordinate, ψ(t) =ωt, with ω being a constant parameter. This yields the following desiredtrajectories for the flat outputs

x∗(t) = R cos(3ωt) cos(ωt), y∗(t) = R cos(3ωt) sin(ωt)

The design parameters for the digital computer simulations were set tobe: λ1 = λ2 = 1, W1 = W2 = 10, ε1 = ε2 = 0.1, R = 4, ω = 0.2.

2.3 Flatness of time-invariant, linear, multivariable systems

Consider the multivariable controllable linear system

x = Ax+Bu, x ∈ IRn, u ∈ IRm


Fig. 1. Closed loop performance for trajectory tracking of a high-gain controllednon-holonomic two wheeled car

where the matrix B is full rank m and constituted by the column vectorsB = [b1, ..., bm]. The system being controllable implies that the n × nm,Kalman controllability matrix

KC =[B,AB, ..., An−1B

]is full rank n.

Controllability of the system implies then that we may extract the follow-ing full rank n× n matrix, C, from the Kalman controllability matrix,

C =[b1, Ab1, ..., A

γ1−1b1, b2, Ab2, ..., Aγ2−1b2, ..., bm, Abm..., Aγm−1bm

]with γi, i = 1, ...,m, being the Kronecker controllability indices of the system,which, evidently, satisfy:

∑i γi = n. In the construction of C we assume that

a column vector of the form Aγj bj , for any j, is eliminated from the collectionwhenever Aγj bj ∈ range KC .

Under the above assumptions, the flat outputs are given by the followingm quantities

F =

φ1

...φm

C−1x

with φi, i = 1, ...,m being n-dimensional row vectors of the form φj =

[0, ..., 0, ...0, 1, 0, ..., 0] with the 1 in the(∑j

i=1 γi

)-th position.


The proof of this fact is quite straightforward by considering the non-singular state coordinate transformation, z = C−1x, which results in thefollowing set of decoupled sub-systems.

zj1 = αjj1 zjγj

+(∑m

i=1,i =j αij1 ziγi

)+ uj

zj2 = zj1 + αjj2 zjγj

+(∑m

i=1,i =j αij2 ziγi

)...zjγj

= zj(γj−1) + αjjγj

zjγj+(∑m

i=1,i =j αijγj

ziγi

)j = 1, 2, ...,m

It should be clear that all the state variables in the j-th subsystem and thecontrol input, uj , can be differentially parameterized by the state variableszjγj

, j = 1, ...,m. These are precisely the components gathered in the vectorF .

Note that for a single-input controllable linear systems, a flat output F isgiven by the linear combination, or any constant multiple thereof, of the orig-inal states obtained by using the last row of the inverse of the controllabilitymatrix.

Example 2. Slding mode control of the linearized PVTOL exampleConsider the normalized nonlinear multivariable system model represent-

ing the Planar Vertical Take-Off and Landing aircraft or better known as thePVTOL aircraft shown in Figure 2.

Fig. 2. PVTOL aircraft

The PVTOL model constitutes a typical example of a multivariable non-minimum phase nonlinear system which is linearizable by means of dynamicendogenous feedback.

x = −u1 sin θ + εu2 cos θ


z = u1 cos θ + εu2 sin θ − 1θ = u2

The normalized system model exhibits, as constant equilibrium states,the following values

x = x, z = z, θ = θ = 0, u1 = u1 = 1, u2 = u2 = 0

The tangent linearization of the system around this constant equilibriumpoint is given by:

xδ = −θδ + εu2δ, zδ = u1δ, θδ = u2δ

The controllability matrix of the system is given by,

C = [b1, Ab1, b2, Ab2, ..., A3b2] =

0 0 0 ε 0 10 0 ε 0 1 00 1 0 0 0 01 0 0 0 0 00 0 0 1 0 00 0 1 0 0 0

The control u1 has controllability index, γ1 = 2, while u2 has controllabilityindex, γ2 = 4. The inverse of the controllability matrix is readily found tobe,

C−1 =

0 0 0 1 0 00 0 1 0 0 00 0 0 0 0 10 0 0 0 1 00 1 0 0 0 −ε1 0 0 0 −ε 0

The flat outputs of the linearized system are then given by the linearcombinations of the original states obtained with the second and last row ofthe controllability matrix. i.e.

F1 = zδ, F2 = xδ − εθδ

The differential parameterization of the states and control inputs in terms ofthe flat outputs is obtained, after a few differentiations, as,

xδ = F2 − εF2, zδ = F1, θδ = −F2, u1δ = F1, u2δ = −F(4)2

The control objective is to bring the system to the constant equilibriumposition. This may be achieved by forcing the flat outputs F1 and F2 tozero in an exponentially asymptotic manner. Thus a pair of sliding surfacecoordinate functions may be prescribed as follows

σ1 = F1 + λ1F1 = zδ + λzδ

σ2 = F(3)2 + k3F2 + k2F2 + k1F2

= −θδ − k3θδ + k2(xδ − εθδ) + k1(xδ − εθδ)


with λ1 > 0 and the coefficients k1, k2, k3 chosen so that the polynomial inthe complex variable s, given by

s3 + k3s2 + k2s+ k1

is a Hurwitz polynomial.The discontinuous incremental feedback control laws

u1δ = −W1sign σ1 u2δ = W2sign σ2

locally create a sliding regime on the sliding surfaces σ1 = 0 and σ2 = 0.The sliding mode control to be applied to the nonlinear system is then

obtained as

u1 = 1−W1sign σ1, u2 = W2sign σ2

Fig. 3. Sliding mode controlled response of nonlinear PVTOL model

Figure 3 shows the controlled responses of the nonlinear system to theactions of the designed multivariable sliding mode controller in a typical rest-to-rest stabilization maneuver. The normalized desired equilibrium positionswere set to be: x = 2, z = 2. The only system parameter, ε, was set to be0.5. The sliding mode controller design parameters were chosen as follows:

λ = 1, W1 = 5, W2 = 5,

k3 = 2ξωn + β, k2 = 2ξωnβ + ω2n, k1 = ω2

nβ,

β = 1, ωn = 0.7, ξ = 0.8

This particular choice places the poles of the incremental ideal equivalentdynamics at the same location of the roots of the polynomials: s + λ and(s+ β)(s2 + 2ξωns+ ω2

n).


2.4 Flatness of Linear Time-Varying Uniformly Controllablesystems

Consider the linear SISO time-varying system

x = A(t)x+ b(t)u, x ∈ Rn, u ∈ R

y = c(t)x y ∈ R (2)

We assume the system to be uniformly controllable ( [11]). In other words,the matrix

C(t) = [ b(t), (A(t)− ddt )b(t), . . . , (A(t)− d

dt )(n−1)b(t)

]is uniformly full rank n over a given finite interval of time [t0, t1].

Uniform controllability implies, according to the results of [6], that thesystem is equivalent to a system in Brunovsky canonical form (a chain of nintegrators) after a static change of coordinates and a state-dependent re-definition of the control input. The system is thus differentially flat. The flatoutput, or Brunovsky output, is directly obtained as a time-varying linearcombination of the original states as indicated in the next proposition.

Proposition 1. A flat output of the uniformly controllable system, x =A(t)x + b(t)u, is given by the following time-varying linear combination ofthe components of the state vector x,

F (t) = [0, . . . , 0, 1]T C−1(t)x

Moreover, any uniformly non-zero scalar, time-varying, multiple of this linearcombination is also a flat output.

ProofConsider the non-singular time-varying state coordinate transformation,

z = C−1(t)x

of the system (2). It is easy to see, that this transformation takes the systeminto the following time-varying linear system

z1 = −α1(t)zn + u

z2 = z1 − α2(t)zn

...zn−1 = zn−2 − αn−1(t)zn

zn = zn−1 − αn(t)zn

Evidently, under such circumstances, zn is the flat output since the controlinput u and all the components of the transformed vector z, and consequentlyall the components of the original state vector x, can be differentially param-eterized in terms of the variable zn and a finite number of its time derivatives.


Example 3. Consider the system

x1 = tu, x2 = u

In this case A(t) = 0 and b(t) = [t; 1]T . The controllability matrix and itsinverse matrix are given by

C(t) =[t −11 0

]; C−1(t) =

[0 1−1 t

]

The controllability matrix rank is 2, independently of t and, hence, the systemis uniformly controllable over any finite time interval of time i.e. the systemis totally controllable. The state coordinate transformation, z1 = x2, z2 =−x1 + tx2 yields,

z1 = u, z2 = z1

A flat output is given by z2 = F = −x1 + tx2. The differential parameteriza-tion of the system variables is given by

x1 = −F + tF , x2 = F , u = F

The previous results may be extended to linear time-varying multivariablesystems, in a manner similar to that used in the time-invariant case. Weproceed by working out an example which illustrates the extension.

2.5 Control around a nominal trajectory of the non-holonomictwo-wheeled car

Consider the simpler model of a non-holonomic two wheeled car,

x = v cos θ, y = v sin θ, θ = ω

where x and y denote the position coordinates of the middle point of thewheels’ axis, θ is the orientation angle with respect to the horizontal axis x.The control inputs are represented by the forward velocity v and the turningangular velocity ω.

The system is differentially flat, with flat outputs given by the coordinatesx and y. The differential parameterization of the system variable in terms ofx, y and a finite number of their time deriviatives is given by,

θ = arctan(

y

x

), v =

√x2 + y2, ω =

yx− yx

x2 + y2(3)

This differential parameterization (3) is most useful in obtaining a nom-inal state and input trajectory. For instance, given the time parameterizedexpressions for a circular trajectory on the plane x− y, as follows,

x∗(t) = R cos γt, y∗(t) = R sin γt (4)


we immediately obtain:

θ∗(t) = arctan (− cot γt) =π

2+ γt, v∗(t) = Rγ, ω∗(t) = γ (5)

We define the incremental variables xδ = x − x∗(t), yδ = y − y∗(t) andθδ = θ − θ∗(t). Consider the Jacobian linearization of the non-holonomiccar equations along the given nominal trajectory, (4)-(5). The system in thematrix form:

xδ = A(t)xδ +B(t)uδ (6)

is obtained as: xδ

yδ

θδ

=

0 0 −v∗(t) sin θ∗(t)0 0 v∗(t) cos θ∗(t)0 0 0

xδ

yδ

θδ

+

cos θ

∗(t) 0sin θ∗(t) 0

0 1

[vδ

ωδ

]

The linearized time-varying system is uniformly controllable on any givenfinite time interval (i.e. the system is totally controllable). Indeed, the con-trollability matrix is given by the array of vectors:

C(t) =[b1(t), (A(t)− d

dt)b1(t), b2(t)

]=

cos θ

∗(t) ω∗(t) sin θ∗(t) 0sin θ∗(t) −ω∗(t) cos θ∗(t) 0

0 0 1

The system has controllability indices: γ1 = 2 and γ2 = 1. The inverse of thecontrollability matrix reveals that the incremental flat outputs are given by:

F1δ = cos θ∗(t)xδ + sin θ∗(t)yδ, F2δ = θδ

The differential parameterization of the system incremental state variablesand incremental control input variables, in terms of the flat outputs F1δ, F2δ,is readily obtained to be

xδ =[sin θ∗(t)F1δ + cos θ∗(t)

(F1δ + v∗(t)F2δ

)]

yδ =1

ω∗(t)

[cos θ∗(t)F1δ + sin θ∗(t)

(F1δ + v∗(t)F2δ

)]

θδ = F2δ

vδ =1

ω∗(t)

[F1δ + v∗(t)F2δ + [ω∗(t)]2F1δ + v∗(t)F2δ

]

ωδ = F2δ

Let ξ, ωn and λ be strictly positive constants. The following set of time-varying feedback control laws:

vδ =1

ω∗(t)

[−2ξωnF1δ −

(ω2

n − [ω∗(t)]2)F1δ + v∗(t)F2δ + v∗(t)F2δ

]

ωδ = −λF2δ


impose, on the closed loop incremental system, the following set of decoupledexponentially asymptotically stable responses which cause the incrementalflat outputs to converge to zero.

F1δ + 2ξωnF1δ + ω2nF1δ = 0, F2δ + λF2δ = 0

Figure 4 shows the simulated closed loop responses of the system to theproposed continuous feedback controller.

Fig. 4. Closed loop responses of the two wheeled car system

A controller locally creating stabilizing motions on the intersection ofthe sliding surfaces, can also be synthesized by setting the required slidingsurfaces and the high-gain control policies to be,

σ1δ = F1δ + λ1F1δ, vδ = − W1

ω∗(t)

(σ1δ

|σ1δ|+ ε1

), 0 < ε1 << 1

σ2δ = F2δ, ωδ = −W2

(σ2δ

|σ2δ|+ ε2

), 0 < ε2 << 1

3 Sliding mode control without state measurements

Here we first explore, in the context of linear time-invariant systems, the pos-sibilities of achieving stabilizing sliding motions without state measurements.The technique, which does not use asymptotic state observers either, relieson measurements of available inputs and outputs only. A state-dependentsliding surface, known to create an asymptotically stable motion towards the


desired equilibrium, is synthesized on the basis of inputs, outputs and iter-ated integrals of such quantities. The effect of the unknown initial values ofthe state variables is compensated via integral control actions entitling againiterated integration of the output stabilization error. We briefly develop asimple theoretical basis for such an approach.

3.1 Generalities on Sliding Mode Control of Linear MultivariableSystems

Consider the controllable and observable linear, time-invariant, multivariablesystem, with m inputs and m outputs

x = Ax+Bu, x ∈ IRn, u ∈ IRm

y = Cx, y ∈ IRm (7)

The unknown initial state vector, at time 0, is denoted by x(0).Let S be an m × n, full rank m, constant matrix with rows vectors,

si, i = 1, ...,m, i.e. S = [s1; ...; sm], such that the constant, m × m, matrixSB is full rank m, i.e. it is invertible. Assume that the matrix S is chosen sothat the projection matrix, M = (I − B(SB)−1S), is such that the matrixMA exhibits all its non-zero eigenvalues with strictly negative real parts. Thismeans that when restricted to the the null space of S, (which we denote byN (S) = x : Sx = 0), the dynamics: x = MAx, x ∈ N (S) is asymptoticallyexponentially stable to the zero state.

Consider then, the set of sliding surfaces, represented by the set of equal-ities, σ = Sx = 0, i.e. σi = six, i = 1, ...,m. These sets are fully described bythe null space of S. Let D be a positive definite m × m matrix of the formD = W + ∆ which we may choose to be diagonal with entries of the form:ωi + δi, ωi, δi > 0 for all i. Away from N (S), the time derivative of thevector, σ = Sx, is given by

σ = SAx+ (SB)u

Let SIGN (·) be a vector operator of signum functions, whose i-th compo-nent, (SIGN σ)i is just given by: (SIGN σ)i = sign σi. The choice u =−(SB)−1D SIGN σ, yields

d

dt

(12σTσ

)= σT σ = σTSAx− σT D SIGN σ

= σT (SAx−D SIGN σ)− σT W SIGN σ

If we choose an open neighborhood, Σ, of the state space, containing theorigin and also containing a non-empty intersection with the subspace N (S),where the following relations are satisfied for all i,

δi > |siAx| ∀ x ∈ Σ


then, it follows that, in Σ, the following inequalities are satisfied,

d

dt

(12σTσ

)≤ −σT W SIGN σ = −

m∑i=1

ωi| σi | < 0

As a consequence, a sliding regime is locally achieved on the sliding sur-face, N (S), which asymptotically exponentially stabilizes the motions of thesystem to the origin of the state space.

We remark that the choice: u = −(SB)−1SAx − (SB)−1W SIGN σ,globally achieves a sliding motion on N (S).

3.2 Integral reconstructors of the state

Consider again the linear time-invariant multivariable system (7). Integratingthe system equations we obtain

x(t) = A

∫ t

0

x(σ1)dσ1 +B

∫ t

0

u(σ1)dσ1 + x(0)

Substituting the expression for x, back into the same equation, we obtain:

x(t) = A2

∫ t

0

∫ σ1

0

x(σ2)dσ2dσ1 +AB

∫ t

0

∫ σ1

0

u(σ2)dσ2dσ1

+B

∫ t

0

u(σ1)dσ1 + x0 +A

∫ t

0

x0dσ1

We adopt the following notation for iterated integration(∫ t

0

φ(t)dt)[j]

=∫ t

0

∫ σ1

0

· · ·∫ σj−1

0

φ(σj)dσjdσj−1 . . . dσ1

with(∫ t

0φ(σ)dσ

)[0]

= φ(t).Carrying out the indicated substitution process, repeatedly, we obtain for

some integer ρ satisfying: n > ρ > 1:

x(t) = Aρ−1

(∫ t

0

x(t)dt)[ρ−1]

+ρ−1∑i=1

Ai−1B

(∫ t

0

u(t)dt)[i]

+ρ−2∑i=1

Ai−1

(∫ t

0

x0dt

)[i−1]

= Aρ−1

(∫ t

0

x(t)dt)[ρ−1]

+ρ−1∑i=1

Ai−1B

(∫ t

0

u(t)dt)[i]

+ρ−2∑i=1

Ai−1x0ti−1

(i− 1)!(8)


Consider now the nm-dimensional column vector of successive derivativesof the outputs and its relation with the states and the inputs

yy...

y(n−1)

=

CCA...

CAn−1

x+

0 0 · · · 0CB 0 · · · 0CAB CB · · · 0...

.... . .

...CAn−2B CAn−3B · · · CB

uu...

u(n−2)

Let the matrix C be constituted by m independent row vectors denotedby C = [c1; ...; cm]. The observability of the system implies that the Kalmanobservability matrix KO = [C;CA; ....;CAn−1] is an nm × n matrix of rankn. Observability means that we may choose n independent row vectors of thematrix KO by eliminating those row vectors of the form: cjA

ρj , which are inthe rank of KO for, and beyond, some integer ρj for each j. We obtain:

y1

...y(ρ1−1)1...

ym

...y(ρm−1)m

=

c1...

c1Aρ1−1

...cm

...cmAρm−1

x+M

uu...

u(ρ−2)

, ρ = maxj ρj

where M is a matrix depending on A, B, C and the observability indicesρj . We denote by O the matrix O = [c1; c1A; ...; c1Aρ1−1; ...; cm; ...; cmAρm−1]From the above relations we obtain

x = O−1

y1

...y(ρ1−1)1...

ym

...y(ρm−1)m

−M

uu...

u(ρ−2)

This last relation implies that we may write the quantity,(∫ t

0x(t)dt

)[ρ−1]

,solely in terms of outputs and iterated integrals of inputs and outputs.

(∫ t

0

x(t)dt)[ρ−1]

=ρ−1∑i=0

[Pi

(∫ t

0

y(t))[i]

+Qi

(∫ t

0

u(t))[i+1]

](9)


Substituting the expression (9) in (8) we obtain that x may be written as

x(t) =ρ∑

i=0

[Φi

(∫ t

0

y(t)dt)[i]

+Θi

(∫ t

0

u(t)dt)[i+1]

]

+ρ−2∑i=1

Ai−1x0ti−1

(i− 1)!

For appropriate matrices Φi and Θi.We refer to the following quantity as an integral reconstructor of the state

vector x

x(t) =ρ∑

i=0

[Φi

(∫ t

0

y(t)dt)[i]

+Θi

(∫ t

0

u(t)dt)[i+1]

]

Example 4. Consider the circuit shown in figure 5 consisting of two LC os-cillators controlled by the same external input voltage source.

Fig. 5. two input-coupled oscillators

The system is described by the following set of differential equations

C1dv1

dt= i1, L1

di1dt

= −v1 + v

C2dv2

dt= i2, L2

di2dt

= −v2 + v

Let E is an arbitrary positive constant voltage acting as a per unit ref-erence value. We define new (normalized) state variables as: x1 = v1/E,x2 = (i1/E)

√L1/C1, x3 = v2/E, x4 = (i2/E)

√L2/C2, and we also set the

new input variable as: u = v/E. One, hence, obtains the following set ofsecond order equations for the normalized capacitor voltages, x = x1 andy = x3,

x = −ω21(x− u), y = −ω2

2(y − u)

where ω1 = 1/√

L1C1 and ω2 = 1/√

L2C2.It is clear that the system is controllable if and only if ω1 = ω2. The flat

output, which may also be found by inspection, is given by

F = ω22x− ω2

1y


The system is also found to be observable when the output of the systemis taken to be the flat output F . This means that we can provide, moduloinitial conditions, an integral input-output parameterization of the variablesx, y and x, y.

x = −ω21

[1

ω22(ω

22 − ω2

1)

(F + ω2

2(∫

F )[2])− (∫

u)[2]]

y = −ω22

[1

ω21(ω

22 − ω2

1)

(F + ω2

1(∫

F )[2])− (∫

u)[2]]

x =[

ω41

ω22(ω

22 − ω2

1)

((∫

F ) + ω22(∫

F )[3])− ω4

1(∫

u)[3]]

+ω21(∫

u)

y =[

ω42

ω21(ω

22 − ω2

1)

((∫

F ) + ω21(∫

F )[3])− ω4

2(∫

u)[3]]

+ω22(∫

u)

This parameterization allows a similar parameterization of F = ω22 x − ω2

1 y,F = ω2

1ω22(y − x), F (3) = ω2

1ω22(y − x), which are needed for any flatness

based controller.

3.3 Sliding mode control based on integral reconstructors of thestate

It should be clear from the previous developments that the relation linkingthe integral reconstructor, x, and the actual value of the state, x, is given by,

x(t) = x(t) +ρ−2∑i=1

Ai−1x0ti−1

(i− 1)!

= x(t) +ρ−2∑i=1

Ai−1

(∫ t

0

x0dt

)[i−1]

(10)

Our main concern is how to appropriately compensate for the effects of theunknown initial conditions, x0, when the actual value of the state is replacedby its integral reconstructor in a given state-based feedback controller design.In particular, when a set of sliding surface coordinate functions of the form:σ = Sx, are to be synthesized in terms of the integral reconstructor for thestate x, as σ = Sx.

Consider then the following compensated set of sliding surface coordinatefunctions, synthesized in terms of the integral reconstructor of the state,

σc = Sx+ ξ(t)


This set of sliding functions, in light of (10), is clearly equivalent to thefollowing set of perturbed sliding surface coordinates functions,

σc = Sx+ ξ(t)− S

ρ−2∑i=1

Ai−1x0ti−1

(i− 1)!

= Sx+ ξ(t)−ρ−2∑i=1

SAi−1

(∫ t

0

x0dt

)[i−1]

Let Γi, i = 1, ..., ρ−2 be an n×m matrix of constant entries. We choose ξ tobe given by ξ(t) =

∑ρ−2i=1 SAi−1(

∫ t

0Γiζ(t)dt)[i] with ζ(t) = y(t) and ζ(0) = 0.

The compensated sliding surface coordinate functions are thus expressed as

σc = Sx−ρ−2∑i=1

SAi−1

(∫ t

0

(x0 − Γiζ) dt)[i−1]

The fundamental idea behind GPI control is to appropriately choose thegain matrices, Γi, so as to guarantee that asymptotically, σc → σ for any x0.Certainly, the class of time-varying functions that need to be compensatedin the above scheme, corresponds to those that can be satisfactorily matchedby means of iterated integral control actions based, solely, on the outputsof the system (namely: constants, ramps, parabolic ramps, etc). That this iscertainly possible, in a stable manner, stems from the controllability of theoriginal system and of its associated integral-based extensions. We shall notspecifically demonstrate the generality of this remark here, and, rather, referthe reader to Fliess et al, [4] for a complete theoretical support. We proceedto illustrate the proposed approach by means of several examples.

Example 5. GPI Sliding Mode Control of the PVTOLConsider again the tangent linearization of the PVTOL model around a

given equilibrium point,

xδ = −θδ + εu2δ, zδ = u1δ θδ = u2δ

Suppose that the system incremental outputs are given by

y1δ = xδ y2δ = zδ

We obtain the following integral reconstructor, or integral parameterization,of the states variables of the system, and of the flat outputs F1 = zδ, F2 =xδ − εθδ

θδ = (∫

u2δ)[2], xδ = y1δ, zδ = y2δ

θδ = (

∫u2δ), xδ = −(

∫u2δ)[3] + ε(

∫u2δ), zδ = (

∫u1δ)


F2 = y1δ − ε(∫

u2δ)[2], F1 = y2δ

F 2 = −(∫

u2δ)[3],F 1 = (

∫u1δ)

F 2 = −(∫

u2δ)[2], F(3)2 = −(

∫u2δ)

The exact relations between the integral reconstructors of the flat outputs,and the corresponding reconstructors of their time derivatives, and the actualvalues of such variables are clearly given by

F2 = F2 − εθδ0t− εθδ0, F2 =F 2 − 1

2θδ0t

2 − θδ0t+ xδ0 − εθδ0

F2 =F 2 − θδ0t− θδ0, F

(3)2 = F

(3)2 − θδ0

F1 = F1 = y2δ, F1 =F 1 + zδ0

We recall the following flatness based sliding surfaces used in the previoussection:

σ1 = F1 + λ1F1 = zδ + λzδ

σ2 = F(3)2 + k3F2 + k2F2 + k1F2

= −θδ − k3θδ + k2(xδ − εθδ) + k1(xδ − εθδ)

The corresponding integral reconstructors of the sliding surface coordinatefunctions are thus given by

σ1 =F 1 + λ1F1, σ2 = F

(3)2 + k3

F 2 + k2F 2 + k1F2

Using the above relations we readily obtain the form of the actual relationsbetween the integral reconstructors of the sliding surface coordinate functionsand their actual values.

σ1 = σ1 + zδ0, σ2 = σ2 + a+ bt+ ct2

where a, b and c are unknown constants which depend on the initial conditionsof, θδ, θδ, xδ and xδ.

We thus propose the following integrally compensated sliding surface co-ordinate functions

σ1c = σ1 + γ11

∫ t

0

y2δdτ

=∫ t

0

(u1δ + γ11y2) dτ + λy2

σ2c = σ2 + γ21

∫ t

0

y1δdτ + γ22

∫ t

0

∫ τ

0

y1δdτdρ+ γ23

∫ t

0

∫ τ

0

∫ ρ

0

y1δdτdρdα


= k1y1 +∫ t

0

(γ21y1 − u2δ)dτ +∫ t

0

∫ τ

0

(γ22y1 − (k1ε+ k3)u2δ) dτdρ

+∫ t

0

∫ τ

0

∫ ρ

0

(γ23y1 − k2u2δ) dτdρdα

To show the effectiveness of the proposed integrally compensated slidingsurface coordinate functions, suppose, first, that the sliding mode condition:σ1c = 0, is permanently achieved by some appropriate discontinuous controlaction. Then we have

σ1c = σ1 − zδ0 + γ11

∫ t

0

zδdτ

= F1 + λF1 −(zδ0 − γ11

∫ t

0

zδdτ

)= 0

Taking one time derivative, and using the fact that z1δ = F1, this last ex-pression is seen to be equivalent to the following expression

F1 + λF1 + γ11F1 = 0

Evidently constants λ and γ11 can always be found such that the imposedsliding condition creates an asymptotically stable motion of the variable F1

towards zero.Similarly, setting σ2c = 0 we obtain, after taking three time derivatives

of the resulting equality, and letting y1δ = xδ = F2 − εF2,

F(6)2 + k3F

(5)2 + (k2 − γ21ε)F

(4)2 + (k1 − εγ22)F

(3)2 + (γ21 − εγ23)F2

+γ22F2 + γ23F2 = 0

The Laplace transform of this linear differential equation is a polynomial inthe complex variable s that can be made into a Hurwitz polynomial by appro-priate choice of the constant design parameters k1, k2, k3, and γ21, γ22, γ23.The constrained motions for the flat output F2 can be made asymptoticallyexponentially stable to zero.

Figure 6 shows the performance of the closed loop nonlinear PVTOLsystem to the GPI sliding mode controller, based on the tangent linearization,given by:

u1 = 1−W1 sign σ1c, u2 = W2 sign σ2c

The closed loop decoupled “characteristic polynomial” for F2 was set tobe: (s2+2ξ2ωn2s+ω2

n2)3 with ξ2 = 0.8 and ωn2 = 0.6, while that correspond-

ing to F1 was chosen as : s+ 2ξ1ωn1s+ ω2n1, with ξ1 = 0.8 and ωn1 = 1.


Fig. 6. GPI Sliding mode controlled responses of nonlinear PVTOL model

4 GPI sliding mode control of an Inertia-spring DCmotor system

4.1 The inertia-spring-DC motor system model

Consider the electro-mechanical system shown in Figure 7. The mathematicalmodel of this system is readily obtained as:

LI = −RI − keω + u, Jω = −Bω − kθ + kmI, θ = ω (11)

where I is the DC motor armature circuit current, θ is the angular dis-placement of the motor axis, measured with respect to a fixed but arbitraryreference position, and ω is the corresponding angular velocity of the motoraxis. The control input, denoted by u, represents the external variable voltageapplied to the armature circuit terminals. The parameters L, R, ke represent,respectively, the armature circuit inductance, the circuit resistance and theback electro-motive force constant of the DC motor. The parameters J , B, kand km denote, respectively, the combined rotor and load inertia, the viscousfriction coefficient, the torsion spring coefficient and the DC motor torqueconstant.

Note that the equilibrium of the system, corresponding to a constantdesired value of the inertia load position θ = θ, can be obtained as

y =k

kmθ, u =

Rk

kmθ


Fig. 7. Inertia-spring DC motor system

The system is controllable and, hence, differentially flat, with flat outputgiven by the inertia load, or motor axis, angular position θ. The flat outputsatisfies the following differential polynomial relation with the input,

Jθ(3) +(B +

RJ

L

)θ +(k +

kmke +RB

L

)θ +

Rk

Lθ =

km

Lu

(12)

As it can be easily determined from (11), the system model is observablefor the electrical output y = I. This fact establishes the constructibility of thesystem, which, in turn, implies that all system state variables are parameter-izable in terms of inputs, outputs and iterated integrals of the input and theoutput variables (See Fliess et al [4]). Such an integral input-output param-eterization of the system state variables is given, modulo initial conditions,by

θ = − L

key +

1ke

∫ t

0

[u(τ)−Ry(τ)]dτ

θ = − k

J

∫ t

0

θ(τ)dτ +km

J

∫ t

0

y(τ)dτ − B

Jθ

θ = − k

Jθ +

km

Jy − B

Jθ

I = I = y (13)

The first expression in (13) is obtained by integration of the first equation in(11). The second expression is obtained by integration of the second equationin (11). The third relation is just the second equation in (11). Note that, fornon-zero initial states, the relations linking the actual values of the angularposition derivatives to the structural estimates in (13) are given by

θ = θ + θ0, θ = θ + θ0 − k

Jθ0 t− B

Jθ0

θ = θ − B

Jθ0 +

Bk

J2θ0 t+

(B2

J2− k

J

)θ0 (14)


where θ0 and θ0 denote the initial mass position and the initial mass velocity.We remark that a similar experimental example, dealing with an inertia-

DC-motor system, has been completely worked out by Marquez et al in [7]for the flatness-based regulation of the angular velocity. Their approach usespole placement and it also requires no mechanical sensors.

4.2 GPI Sliding mode control

A sliding surface coordinate function, σ, that ideally induces, by the slidinginvariance condition σ = 0, an asymptotic exponential stabilization of themass position, θ, towards a constant desired value, θ, is given by,

σ = θ + k3θ + k2(θ − θ)

for suitable (Hurwitz) choices of k3 and k2. We propose, nevertheless, thefollowing modified sliding surface coordinate function, which does not usethe otherwise required measurements of the angular position, angular velocityand angular acceleration, but instead uses the structural estimates of thesevariables, previously defined in (13).

σ = θ + k3θ + k2(θ − θ) + k1ξ + k0η (15)

with

ξ = y − k

kmθ, ξ(0) = 0; η = ξ, η(0) = 0

The added iterated integral control action suitably compensates the con-stant and the linearly growing structural estimate errors with respect to theactual values of the flat output and its time derivatives. The underlyingequivalent (but never used) expression of the sliding surface in terms of theactual (unmeasured) values of the load angular position variables, and itstime derivatives, is obtained by replacing (14) into (15). The obtained ex-pression is of the form:

σ = θ + k3θ + k2(θ − θ) +(k1

∫ t

0

[y − k

kmθ]dτ − α

)

+∫ t

0

[k0

∫ τ

0

(y − k

kmθ)dρ− β

]dτ

where the constant parameters α and β depend on the initial conditions forθ and θ. The ideal sliding condition, σ = 0, is easily seen to be equivalent tothe following forth order closed loop system, which is completely independentof any initial condition values.

θ(4) + (k3 + k1J

km)θ(3) + (k2 +

k1B

km+

k0J

km)θ

+(

k1k

km+

k0B

km

)θ +

k0k

km(θ − θ) = 0


where, as before, use has been made of the flatness-based differential param-eterization linking the system output, y = I, to the flat output θ. It is clearthat a suitable choice of the design parameter set k3, k2, k1, k0 yields an ex-ponentially asymptotically stable closed loop dynamics for the mass positionθ.

We propose the following discontinuous sliding mode feedback controller:

u = u−W sign σ, W > 0

4.3 Experimental results

The desired objective was to stabilize the system motions to θ = 0 [rad]from an initial position of θ0 = 0.03 [rad] and an initial current I(0) = 0.183[Amp]. The design constants were chosen to be the coefficients of a fourthorder polynomial, in the complex variable s, of the form: (s2 +2ζωns+ω2

n)2,

with ζ = 0.707, ωn = 80. We also set, W = 10. Figure 8 depicts the systemvariables responses obtained from the used experimental set up.

Fig. 8. Experimental results for the sliding mode based GPI controller

5 Conclusions and Suggestions for Future Work

In this chapter, we have studied in a rather tutorial fashion the implicationsof differential flatness in the design of sliding mode control of dynamic sys-tems. The flatness property was seen to largely trivialize the sliding mode


controller design problem in several classes of dynamic systems. Linear, non-linear, SISO or Multivariable, time-invariant or time-varying systems, all canbe easily treated thanks to the flatness property. We have also explored thepossibilities of incorporating the ideas of Generalized PI controllers into thesliding surface coordinate function synthesis task. This implies, at least inthe linear case, that for observable, or constructible, systems there is no realneed to measure the state vector of the system, nor to build asymptotic stateobservers, in order to effectively synthesize a sliding surface with desired,closed loop, stability features. As demonstrated by several examples, integralreconstructors, based on the availability of inputs and outputs, can be usedfor the sliding surface coordinate function synthesis. The basic advantage ofGPI-based sliding mode control over traditional observer-based sliding sur-face synthesis lies in the enhanced robustness of the integral reconstructors,with their associated integral output error compensation loops, to suddenparameter variations and other external un-modeled uncertainties (see [4] fora preliminary example).

Several important issues deserve attention and further development withinthe context of flatness and integral reconstructor-based sliding surface coor-dinate functions synthesis. One of them is to extend the presented approachesto deal with a significant class of nonlinear SISO and nonlinear multivariablesystems. A second feature is to analyze the central issue of robustness in con-nection with integral reconstructor-based sliding surfaces when the systemis subject to various kinds of uncertainties. A final topic for further study isconstituted by a needed extension of some of these kind of controller synthesisresults to the case of linear delay differential systems.

References

1. Cartan, E. (1953) “Sur l’integration de certains systemes indeterminesd’equations differentielles ” in CEuvres Completes, pp. 1169-1174. Gauthier-Villars.

2. Fliess, M., Marquez, R., and Delaleau,E., (2000) “State feedbacks without as-symptotic observers and generalized PID regulators” Nonlinear Control in theYear 2000, A. Isidori, F. Lamnabhi-Lagarrigue, W. Respondek, Lecture Notesin Control and Information Sciences, (258), 367-384, Springer, London.

3. Fliess, M. (2000) “Sur des pensers nouveaux faisons des vers anciens”.InActes Conference Internationale Francophone d’Automatique (CIFA-2000), 26-36, Lille. France.

4. Fliess, M., Marquez, R., Delaleau, E., and Sira-Ramirez, H., (2001) “CorrecteursPID Generalises et Reconstructeurs Integraux, ” ESAIM: Control, Optimisationand Calculus of Variations (accepted for publication, to appear)

5. Fliess, M.. Levine, J., Martin, P., and Rouchon, P., (1995) “Flatness and defectof nonlinear systems: Introductory theory and examples” International Journalof Control, 61,6, 1327-1361.

6. Malrait, F., Martın Ph., and Rouchon, P. (2001) “Dynamic Feedback Transfor-mations of Controllable Linear Time-Varying Systems” in Nonlinear Control in


the Year 2000, A. Isidori, F. Lamnabhi-Lagarrigue, W. Respondek, (Eds), Lec-ture Notes in Control and Information Sciences, Vol. 259, pp. 55-62, Springer,London.

7. Marquez, R., Delaleau, E., and Fliess, M., (2000) “Commande par PID generalised’un moteur electrique sans capteur mecanique. In Actes Conference Interna-tionale Francophone d’Automatique (CIFA-2000), 453-458, Lille. France.

8. van Nieuwstadt, M. (1996) “Trajectory Generation for Nonlinear Control Sys-tems,” California Institute of Technology, PhD Thesis.

9. Pomet, J. B. (1994) “A Differential Geometric Setting for Dynamic Equivalenceand Dynamic Linearization” INRIA Report No. 2312, Sophia Antipolis, France.

10. Rathinam, M. (1997) “ Differentially Flat Nonlinear Control Systems”, Califor-nia Institute of Technology, PhD Thesis, Pasadena, California. (Also TechnicalReport CDS-97-008, 1997).

11. Silverman, L. M. and Meadows, H. E. (1967) “Controllability and Observabilityin Time-Variable Linear Systems, ” SIAM J. on Control, 5, pp. 64-73.

12. Silverman, L. M. (1966) “Transformation of Time-Variable Systems to Canon-ical (Phase-Variable) Form” IEEE Transactions on Automatic Control, AC-11,pp. 300-303.

13. Sluis, W. M. (1990) “A Necessary Condition for Dynamic Feedback Lineariza-tion” Systems and Control Letters , 15, pp. 35-39.

14. Utkin, V. I. (1977) Sliding mode control in the theory of variable structuresystems, MIR Publishers, Moscow.

On Robust VSS Nonlinear ServomechanismProblem

Vadim Utkin1, B. Castillo-Toledo2, A. Loukianov2, and O.Espinosa-Guerra2

1 Department of Electrical Engineering, Ohio-State University, Columbus, Ohio,43210-1272, USA.

2 Centro de Investigacion y de Estudios Avanzados del IPN, A.P. 31-438,C.P.44550, Guadalajara, Jal, Mexico. toledo[louk]@gdl.cinvestav.mx

Abstract. Analogously to the formulation of the so-called classical servomecha-nism problem, the problem of tracking a reference signal while rejecting the effectof a disturbance signal by means of the VSS technique is studied by formulatingthe sliding mode servomechanism problem for which conditions for the existenceof a solution for in general case and for a classes of nonlinear system presented inthe Regular Form or in the Nonlinear Block Controllable Form are derived. Theeffectiveness of the proposed method is demonstrated by the application to thePendubot system.

1 Introduction

This chapter deals with the problem of controlling the output of a uncer-tain nonlinear system so as to achieve asymptotic tracking of a reference andrejection of unknown admissible disturbances is one of interesting and at-tractive problem in classical control theory. This problem known as robustoutput regulation or robust servomechanism problem, has been studied forthe nonlinear case (see for example, [2,4–6]), for which a continuous regulatorhas been proposed.

In this work, a discontinuous regulator is investigated by combining thesliding mode control [10] and block control technique [7], [8]. The underlyingidea is to design a sliding manifold on which the dynamics of the systemis constrained to evolve by means of a discontinuous control law, instead ofdesigning a continuous stabilizing feedback, as in the case of the classical reg-ulator problem. The sliding manifold contains the steady-state manifold, andthe dynamics of the systems tends asymptotically, along the sliding manifold,to steady-state behavior.

First, the Sliding Mode Servomechanism Problem is formulated, and theconditions of solution of this problem are investigated. Then a discontinuousregulator is designed for a class of nonlinear systems presented in the so-called Regular Form. Finally, we assume that the class of nonlinear systemsto be considered, might be presented in the Nonlinear Block ControllableForm (NBC-form) with uncertainty, consisting of a set of blocks for which


the block state vector and the block fictitious control vector have the samedimension. Such kind of representation enables a reduction of the originalcontrol law synthesis problem into a sequence of lower-order subproblemswhich can be solved applying the block control technique. As a result, thenominal sliding mode dynamics can be linearized, and then discontinuousfeedback can be used to compensate the matched uncertainty. The effec-tiveness of the proposed method is demonstrated by the application to thewell-known Pendubot system.

2 Problem Statement

Consider the following nonlinear plant subject to perturbation:

x = f(x, u, w) +∆f(x, v), x(0) = x0 (1)y = h(x) (2)

with the plant state x, defined in a neighborhood X of the origin of n,the m-dimensional plant control input u, the p-dimensional plant regulatedoutput y, and w is the exogenous signal defined in a neighborhood W of theorigin in q, representing the reference input (to be tracked) produced by anexosystem described by

w = s(w), w(0) = w0, (3)

The unknown vector ∆f represents the model uncertainties, and v is avector of external disturbances (to be rejected). The output tracking error isdefined as

e = y − yref , yref = q(w) (4)

It is assumed that f(x, u, w), s(w), h(x) and q(w) are smooth functionsand f(0, 0, 0) = 0, s(0) = 0, h(0) = 0 and q(0) = 0.

The systems (1), (2), (4) and (3) are characterized by the following as-sumptions:

H1) The pairf,B has a stabilizable linear approximation at x = 0.H2) The Jacobian matrix S = ∂s

∂w (0) at the equilibrium point w = 0 has alleigenvalues on the imaginary axis.

The Classical State Servomechanism Problem (CSSP) [6] in absence of theuncertainty term ∆f , consists on finding a continuous feedback u = α(x,w)that provides the following requirement:

CS) The system x = f(x, α(x, 0), 0) has a locally exponentially stable equi-librium point at the origin x = 0,

CE) There exists a neighborhood U ⊂ X × W such that for each initialcondition (x0, w0) ∈ U the output tracking error e(t) goes asymptoticallyto zero, i.e. limt→∞[y(t)− yref (t))] = 0

344 V. Utkin et al.

In [6] it was shown that the solvability of CSSP can be stated in terms ofthe existence of a pair of mappings x = π(w) and c(w) locally defined in Wwith π(0) = 0 and c(0) = 0, that solve the partial differential equation

f(π(w), u(π(w), w) =∂π(w)

∂ws(w) (5)

h(π(w))− q(w) = 0 (6)

A continuous controller solving the servomechanism problem was obtainedby choosing

u = c(w) +K(x − π(w)) (7)

where K is a matrix which places the eigenvalues of the linear approximationof the closed-loop system (1) and (7) at the equilibrium point x = 0, namely(A+BK) in C−.

3 Sliding Mode Servomechanism Problem

Analogously to the CSSP, we may define the Sliding Mode ServomechanismProblem (SMSP) as the problem of finding a manifold

σ(x,w) = 0, σ ∈ m (8)

and a discontinuous controller

ui =

u+i (x,w) σi(x,w) > 0

u−i (x,w) σi(x,w) < 0

, i = 1, ...,m (9)

where maps u+i (x,w), u−

i (x,w) and σi(x,w) are chosen to induce localasymptotic convergence of the state vector to the switching surface σi(x,w)= 0, i = 1, ...,m so that the following conditions hold:SS). The equilibrium point x = 0 of the closed-loop system (1), (9) and

(8), is locally asymptotically stable;SE). The output tracking error goes asymptotically to zero, i.e.

limt→∞[y(t)− yref (t)] = 0

Note, that the conditions for sliding motion to occur on σi(x,w) = 0 maybe stated in numerous ways. We need

limσi→0+

σi < 0

and

limσi→0−

σi > 0

in the neighborhood of σi(x,w) = 0, i = 1, ...,m, [10].

345On Robust VSS Nonlinear Servomechanism Problem

4 Nonlinear Systems Affine in Control

In this section we consider the nonlinear systems that is linear in control andperturbation inputs, in the absence of the uncertainty term, that is

x = f(x) +B(x)u+D(x)w (10)

Let us first investigate the conditions of the solution of SMSP assumingthat the initial state vector of the system (10) in the manifold σ(x,w) = 0(8), and the sliding mode occurs with the state trajectories confined to thismanifold for t > 0.The projection motion of the system on the subspace σ isdescribed by the m order system

σ = G(x,w)f(x) +G(x,w)B(x)u+G(x,w)D(x)w +H(x,w)s(w) (11)

where G = ∂σ∂x , H = ∂σ

∂w .

Following the equivalent control method [10] we put σ = 0, that is

G(x,w)f(x) +G(x,w)B(x)u+G(x,w)D(x)w +H(x,w)s(w) = 0

Assuming that the matrix (G(x,w)B(x)) is nonsingular for any x,w ∈ U,we may find the solution of this algebraic equation, that is, the equivalentcontrol, ueq as

ueq = −(G(x,w)B(x))−1[G(x,w)f(x) +G(x,w)D(x)w +H(x,w)s(w)]

and substitute it for discontinuous control u in the original system (10):

x = P (x,w)[f(x) +D(x)w]− M(x,w)H(x,w)s(w) (12)

where P (x,w) = In − M(x,w)G(x,w), M(x,w) = B(x)(G(x,w)B(x))−1.

The systems (12) and (3) can be represented as

x = PAx+ [PD − B(ΣB)−1HS]w + ψ1(x,w) (13)w = Sw + ψ2(w) (14)

where P1 =[In − B(ΣB)−1Σ

], A = ∂f

∂x (0), B = B(0), D = D(0), Σ =G(0, 0), H = H(0, 0) and S = ∂s

∂w (0) with functions ψ1(x,w) and ψ2(w)vanishing at the origin with its first derivatives. Assume now that the matrix(PA) is stable, and by assumption H2 the eigenvalues of the matrix S areon the imaginary axis. Therefore, the system (13) and (14) has a centralmanifold x = π(w) at (0, 0) with Ck mapping π(w) , (π(0) = 0) satisfyingthe condition

P (π(w), w)[f(π(w)) +D(π(w))w] = M(π(w), w)H(π(w), w)s(w) (15)

From this it is possible to derive a condition for solution to the SMSP.

346 V. Utkin et al.

Proposition 1. Under assumptions H1 and H2, if there exist Ck ( k ≥2) mapping π(w) with π(0) = 0 which satisfies

P (π(w), w)[f(π(w)) +D(π(w))w] = M(π(w), w)Σ∂π

∂ws(w) (16)

h(π(w))− q(w) = 0 (17)

Then the SMSP is solvable.Proof. Proceeding along the previous discussion, setting

σ = Σz, z = x − π(w), (18)

then conditions (15) and (16) are identical. Choosing

u = −k(G(x,w)B(x))−1sign(σ) + ueq(x,w) (19)

ueq(x,w) = −[ΣB(x)]−1[Σf(x) +ΣD(x)w +Σ∂π

∂ws(w)] (20)

the control action (19) with (20) and k > 0 guarantees a sliding mode motionon the surface σ = 0 (18) that described by (12). The Jacobian matrix of[P (x,w)f(x)] in the (12) equal to matrix (PA), and by assumption H1 thereexists matrix Σ such that (n − m) eigenvalues of matrix (PA) are in C−.Therefore, by a property of center manifolds, under condition (15) or (16),we have on the sliding manifold σ = 0 (18) in steady state

PAz = 0 (21)Σz = 0 (22)

It is straightforward to verify that P is a projection operator along the rangespace of B onto the null space of Σ [1] i.e.

PB = (In − B(ΣB)−1Σ)B = 0 (23)

and

Pz = z ∀z ∈ N , N = z ∈ Rn | Σz = 0 (24)

Therefore by conditions (21)-(23) we have z = 0 that means x = π(w).Thus the first requirement SS) is fulfilled. So, by continuity, if condition (17)holds, then the second requirement SE) is also fulfilled.

The later result provides an existence condition for deriving a slidingmanifold on which the output tracking error is zeroed. In contrast to theclassical problem it does not need to calculate the steady state value fordiscontinuous control. Indeed, the equivalent control in sliding mode tendsto its steady state

ueq(π(w), w) = −[ΣB(π(w))]−1[Σf(π(w)) +ΣD(π(w))w +Σ∂π

∂ws(w)]

automatically.The obtained result can be viewed more clearly in the case of nonlinear

systems in regular form.


5 Nonlinear Systems in Regular Form

Consider the nonlinear system (10), (2) and (4), and assume that the system(10), by a diffeomorphism x′ = ϕ1(x) can be transformed in the regular form[9]

x′1 = f1(x′

1, x′2) +D1(x′

1, x′2)w (25)

x′2 = f2(x′

1, x2) +B2(x′1, x

′2)u+D2(x′

1, x′2)w (26)

where x′ = (x′1, x

′2)

T , x′1 ∈ n−m, x′

2 ∈ m and rankB2(x′1, x

′2) = m ∀x ∈ n.

Let the sliding manifold (8) be described by

σ = x′2 + σ1(x′

1, w) = 0 (27)

where σ1(x1, w) is a Ck ( k ≥ 2) mapping with σ1(0, 0) = 0, and setting

u = −kB−12 (x′

1, x′2)sign(σ) + ueq (28)

ueq = −B−12 (x′

1, x′2)

[f2(x′

1, x′2) +

∂σ1

∂x′1

f1(x′1, x

′2) (29)

+(

D2(x′1, x

′2) +

∂σ1

∂x′1

D1(x′1, x

′2)

)w +

∂σ1

∂ws(w)

](30)

then the control (28) and (29) guarantees a sliding motion on σ = 0 given by(27) in a finite time. On this surface, the dynamics of the closed-loop system(25), (26) and (28) are given by the reduced order equation

x′1 = f1(x′

1, σ1(x′1, w)) +D1(x′

1, σ1(x′1, w))w (31)

e = h′(x′1, σ1(x′

1, w))− q(w) (32)

with h′(x′1, x

′2) = h(ϕ−1

1 (x)). At this point, the system (31) and (3) can berepresented as

x′1 = (A11 +A12Σ1)x′

1 + (A12L1 +D1)w + ψ1(x′1, w) (33)

w = Sw + ψ2(w) (34)

where A11 = ∂f1∂x′

1(0, 0), A12 = ∂f1

∂x′2(0, 0), D1 = D1(0), Σ1 = ∂σ1

∂x′1(0, 0), L1 =

∂σ1∂w (0, 0) and S = ∂s

∂w (0) , with functions ψ1(x′1, w) and ψ2(w) vanishing at

the origin with its first derivatives. Under assumption H1 there exists matrixΣ1 such that matrix (A11 +A12Σ1) is Hurwitz. Therefore the system (33)and (34) under assumption H2 has a central manifold x′

1 = π1(w) at (0, 0)with Ck mapping π1(w) , (π1(0) = 0) satisfying the condition

∂π1

∂ws(w) = f1(π1(w), σ1(π1(w), w)) +D1(π1(w), σ1(π1(w), w))w. (35)

From this it is possible to deduce a condition for the solution slidingmode servomechanism problem for a nonlinear system in regular form (25)and (26).

348 V. Utkin et al.

Proposition 2. Under assumptions H1 and H2, if there exist C k ( k ≥2) mappings c1(w) and π1(w) with c1(0) = 0 and π1(0) = 0 which satisfy

f1(π1(w), c1(w)) +D1(π1(w), c1(w))w =∂π1

∂ws(w) (36)

h′(π1(w), c1(w))− q(w) = 0 (37)

Then the SMSP is solvable.Proof. Proceeding along the previous discussion, setting

σ1(x′1, w) = c1(w) +Σ1(x′

1 − π1(w)) (38)

then the Jacobian matrix of f1(x′1, σ1(x′

1, 0)) in the (31) equal to matrix(A11 +A12Σ1) which, by assumption H1, can be made stable by appropriatechoice of Σ1. Therefore, by a property of center manifolds, x′

1(t) → π1(w(t)),and thus σ1(π1(w), w) = c1(w), in the manifold x′

1(t) = π1(w(t)), and condi-tion (36) reduces to condition (35), so, by continuity, if condition (37) holds,then the output tracking error converges to zero.

The proposed controller (28) and (29) with (27) and (38) provides onlylocal stability of the equilibrium point x = 0. In the next section we considera globally stabilizing discontinuous feedback.

6 Nonlinear System in NBC-Form

In this section a discontinuous recontrol strategy will be investigated for aclass of nonlinear system (1) linear with respect to inputs signal

x = f(x) +B(x)u+D(x)w + F (x)v (39)

where ∆f(x, v) = F (x)v, using the block control linearizing technique [8].The essential feature of the proposed method is the transformation of equa-tion (39) to the NBC-form consisting of r blocks:

xr = fr(xr) +Br(xr)xr−1 +Dr(xr)w + Fr(xr)v (40)xi = fi(xi, ..., xr) +Bi(xi, ..., xr)x

+Di(xi, ..., xr)wi−1 + Fi(xi, ..., xr)v, i = 2, ..., r − 1 (41)x1 = f1(x) +B1(x)u+D1(x)w + F1(x)v (42)

with output error e = h(x)− q(w) where the transformed vector x is decom-posed as x = (x1, x2, ...., xr)T and xi is a ni×1 vector. In each block, the vec-tors xi−1 are regarded as fictitious control vectors, and rankBi = ni,∀x ∈ X.The integers (n1, n2, ..., nr) define the plant structure, and

∑ri=1 ni = n.


6.1 Block Decomposition of Nonlinear System

The procedure of reducing the system (39) to the NBC-form (??) consistsof a series of steps. The main technique is the transformation of an affinecontrol system into the regular form by means of the integral surface method[9].Step 1. Assume that rank B(x, t) = n1 ≤ m , ∀x ∈ X, and, possibly

after reordering, there exists an (n1 × m) block B1(x1, x12) such that

B(x) =[B12(x1, x12)B1(x1, x12)

], f(x) =

[f12(x1, x12)f1(x1, x12)

],

with rank B1(x1, x12) = n1, ∀x ∈ X, where x1 and x12 are n1 × 1 and(n − n1) × 1 vectors, respectively. At this point, we introduce the followinginstrumental assumptions which will be carried for each step of the procedure.A11) The Pfaffian system

dx12 +A1(x)dx1 = 0, A1(x) = −B12(x)B+1 (x) (43)

is completely integrable, that is the condition

∂aiα

∂xβ−

n−n1∑j=1

aiα

∂aiα

∂xj=

∂aiβ

∂xβ−

n−n1∑j=1

aiβ

∂aiβ

∂xj

where A1(x) = aiα(x), i = 1, ..., n − n1, α and β ∈ (n − n1 + 1), ..., n ,

and B+1 is the right pseudo inverse of B1 holds.

A12) The mappings F (x) and D(x) can be decomposed as

D(x) = Dm(x1, x12) +Du(x12)F (x) = Fm(x1, x12) + Fu(x12)

where Fm(x) and Dm(x) satisfy the matching conditions, namely

Fm(x),Dm(x) ∈ span /B(x)

Under assumption A11, it is possible to show that a solution of the equa-tion (43) is given by x12 = ϕ1(x1, c) where c = (c1, c2, ...., cn−n1)

T is an in-tegration constants vector. From this solution, the vector c can be obtained,namely, c = ϕ1(x1, x12), and taken as a local change of coordinates given byx′

2 = ϕ1(x1, x12).Now, using assumption A12, and partitioning D and F as

D =(

D12(x)D1(x)

)=

(Dm

12(x1, x12) +Du12(x12)

Dm1 (x1, x12) +Du

1 (x12)

)

F =(

F12(x)F1(x)

)=

(Fm

12(x1, x12) + Fu12(x12)

Fm1 (x1, x12) + Fu

1 (x12)

)

350 V. Utkin et al.

it is possible to show (see appendix) that the original system (39) under thenew coordinates is described by

x′2 = f ′

2(x1, x′2) +D′

2(x′2)w + F ′

2(x′2)v (44)

x1 = f1(x1, x′2) +B1(x1, x

′2)u+D1(x1, x

′2)w + F1(x1, x

′2)v (45)

Note that in the terminology of [3], equation (45) exhibits a controlled dy-namics, and (44) exhibits an uncontrolled one with fictitious control inputx1. The following assumption is fundamental to derive the NBC-form:A13) The mapping f ′

2(x1, x′2) in system (44) is affine in its first argument,

namely

f ′2(x1, x

′2) = f ′′

2 (x′2) +B′

2(x′2)x1 (46)

At this point, we may consider three cases:

a) rank B′2(x

′2) = 0, ∀x′

2 ∈ X. This is equivalent to having an uncontrollablesystem. For the purposes of this work, we will assume in the followingthat the original system is controllable.

b) rank B′2(x

′2) = n − n1 ∀x′

2 ∈ X. In this case, after defining x2 = x′2,

B2 = B′2, f2 = f ′

2, D2 = D′2, F2 = F ′

2, the NBC-form is:

x2 = f2(x2) +B2(x2)x1 +D2(x2)w + F2(x2)vx1 = f1(x1, x2) +B1(x1, x2)u+D1(x1, x2)w + F1(x1, x2)v

c) rank B′2(x

′2) = n2 < n − n1. In this case, a subsequent step is necessary,

and the subsystem (44) and (46) with state x′2 and input x1 is further

decomposed and transformed similarly to Step 1.

Step k. Consider the system obtained at (k − 1)th step

x′k = f ′

k(x′k) +B′

k(x′k)xk−1 +D′

k(x′k)w + F ′

k(x′k)v (47)

xi = fi(xi, ..., x′k) +Bi(xi, ..., x

′k)xi−1

+Di(xi, ..., x′k)w + Fi(xi, ..., x

′k)v, i = 2, ..., k − 1 (48)

x1 = f1(x1, ..., x′k) +B1(x1, ..., x

′k)xr−1

+D1(x1, ..., x′k)w + F1(x1, ..., x

′k)v (49)

where rank Bi(xi, ..., x′k) = ni, i = 1, ..., k − 1. If nk =rank Bk(x′

k) is suchthat nk = n − ∑k−1

j=1 nj , we define xk = x′k, fk = f ′

k, Dk = D′k, Fk = F ′

k,and the algorithm terminates giving the equations (47)-(49) as the desiredNBC-form. If nk < n − ∑k−1

j=1 nj , the subsystem (47) is partitioned as

xk2 = fk2(xk, xk2) +Bk2(xk, xk2)xk−1 +Dk2(xk, xk2, )w + Fk2(xk, xk2)vxk = fk(xk, xk2) +Bk(xk, xk2)xk−1 +Dk(xk, xk2)w + Fk(xk, xk2)v

where rank Bk = nk, x′k = (xk, xk2)T , and xk and xk2 are nk × 1 and

(n − ∑k−1j=1 nj − nk) × 1 vectors, respectively. For this step, we generalize

assumptions A11 and A12 as follows:


Ak1) The Pfaffian system

dxk2 − Bk2B+k dxk = 0 (50)

is completely integrable.Ak2) The mappings F ′

k and D′k can be decomposed in the form

D′k(x

′k) = Dm

k (xk, xk2) +Duk (xk2)

F ′k(x

′k) = Fm

k (xk, xk2, v) + Fuk (xk2)

where Fmk and Dm

k satisfy the matching conditions, namely

Fkm,Dk

m ∈ spanBk′(x′

k)

Proceeding as in the first step, under the previous assumptions, we mayfind a local change of coordinates given by x′

k+1 = ϕk(xk, xk2) where ϕk iscomputed from the solution of (50) such that the system is described by

x′k+1 = f ′

k+1(xk, x′k+1) +D′

k+1(x′k+1)w + F ′

k+1(x′k+1)v

xk = fk(xk, x′k+1) +Bk(xk, x′

k+1)xk−1

+Dk(xk, x′k+1)w + Fk(xk, x′

k+1)vxi = fi(xi, xi+1, ..., x

′k+1) +Bi(xi, xi+1, ..., x

′k+1)xi−1

+Di(xi, xi+1, ..., x′k+1)w + Fi(xi, xi+1, ..., x

′k+1)v, i = 2, ..., k − 1

x1 = f1(x1, x2, ..., x′k+1) +B1(x1, x2, ..., x

′k+1)xr−1

+D1(x1, x2, ..., x′k+1)w + F1(x1, x2, ..., x

′k+1)v

with rankBi = ni, i = 1, ..., k.In the same way, assumption A13 for step k is stated as:Ak3) The mapping f ′

k+1(xk, x′k+1) is affine in its first argument, namely

f ′k+1(xk, x′

k+1) = f ′k+1(x

′k+1) +B′

k+1(x′k+1)xk

From the previous algorithm, we may state the following result:Theorem 1. Assume that the system (39) is controllable and at each step ofthe NCB-form algorithm assumptions Ak1, Ak2 and Ak3 hold. Then, thereexists an integer r ≤ n such that the system (39) takes the form (40)–(42).

6.2 Block Linearization of Nominal System

In this section, following the block control design technique [8], a nonlin-ear transformation linearizing the nominal part of uncontrolled dynamics isderived. Based on this transformation, a nonlinear sliding manifold will beproposed in the next section.

At this point we introduce the following assumptionH3) The uncertainty term ∆f(x)v in (1) satisfies the matching condition.

If condition H3 holds, then in (40)–(41)

352 V. Utkin et al.

Fr(x1)v = 0Fi(xi, xi+1, ..., xr)v = 0, i = 2, ..., r − 1

It is more convenient to renumber the state variables of (40)–(41) in thereverse form and represent this system as

x1 = f1(x1) +B1(x1)x2 +D1(x1)w (51)x2 = f2(x2) +B1(x2)x3 +D2(x2)w (52)xi = fi(xi) +Bi(xi)xi+1 +Di(xi)w, i = 3, ..., r − 1 (53)xr = fr(x) +Br(x)u++Dr(x)w + Fr(x)v (54)

with output error e = h(x) − q(w) where xi = (x1, ..., xi)T , i = 2, ..., r − 1.The integers (n1, n2, ..., nr) satisfy

n1 ≤ n2 ≤ · · · ≤ nr ≤ m. (55)

The condition ni−1 ≤ ni (55) means ni−1 = ni or ni−1 < ni. In thesequel we consider the system (51)–(54) with the structure

n1 = n2 < n2 · ·· < nr = m (56)

that includes both of the cases.A nonlinear linearizing state transformation can be derived considering

the state xi+1, i = 1, ..., r − 1 as a fictitious control vector in the ith block.This procedure is outlined in the following steps.Step1. Since n1 = n2, the matrix B1(x1) is square and the inverse matrix

B−11 (x1) exists. Setting z1 = x1 := α1(x1), the fictitious control x2 in (51)

can be chosen as

x2 = xc2(x1, w) +B−1

1 (x1)(K1z1 + z2), (57)

where z2 is n2 × 1 vector of new variables, K1 is a matrix with desiredeigenvalues, and xc

2(x1, w) is calculated from the equation z1 = 0 along thetrajectories of (51), namely,

xc2 = −B−1

1 (x1)[f1(x1) +D1(x1)w] (58)

The transformed 1st block with new coordinates z1, z2 and input (57) and(58) has the form

z1 = K1z1 + z2 (59)

The variable z2 can be obtained using (57) and (58) as

z2 = B1(x1)x2 + f1(x1)− K1α1(x1) +D1(x1)w := α2(x2, w) (60)

Step 2. Taking the derivative of (60) along the trajectories of the system(??) and (3) yields

z2 = f2(x2, w) + B2(x2)x3 (61)


where f2(x2, w) = ∂α2∂x1

[f1(x1)+B1(x1)x2+D1(x1)w]+∂α2∂x2

[f2(x2)+D2(x2)w]+∂α2∂w s(w), B2 = B1B2. Note that rank B2 =rank B2 = n2. Since n2 < n3 thematrix B2 as well as B2 are not square. As on the first step, the fictitiousinput vector x3 in (61) is chosen similar to (57) and (58):

x3 = xc3(x2, w) + B+

2 (x2)[K2z2 + E2,1z3] (62)

where z3 is n3×1 vector, K2 is a desired matrix, B+2 denotes the right pseudo

inverse of B2, E2,1 =[In2 0

], and again xc

3(x2, w) is found from the equationz2 = 0 (61) being

xc3(x2, w) = −B+

2 (x2)f2(x2, w) (63)

Thus, equation (61) with (62) and (63) takes the same form of equation (59),namely

z2 = K2z2 + E2,1z3.

Now, we establish the following assumption.B2) The elements of matrix B2(x2) can be ordered such that the squarematrix

B3(x2) :=[B2(x2)E2,2

]

with E2,2 =[0 In3−n2

], has rank n3.

Based on this assumption, the variable z3 can be obtained using (62), (63)and (60) as

z3 = B3(x2)x3 +[f2(x2, w) +K2α2(x2, w)

0

]:= α3(x3, w)

Step i. At this stage, it is possible to show that if we have, after the (i−1)th step, the transformed blocks of the system (51)–54) with new variablesz1, z2, .., zi−1 (under assumption nj < nj+1, i = 3, ..., r − 1) of the form

z1 = K1z1 + z2 (64)z2 = K2z2 + E2,1z3 (65)

...zi−1 = Ki−1zi−1 + Ei,1zi (66)

with

zi = Bi(xi−1)xi +[fi−1(xi−1, w) +Ki−1αi−1(xi−1, w)

0

]:= αi(xi, w)(67)

Bi(xi−1) :=[Bi−1(xi−1)

Ei−1,2

]; and rank Bi = ni. Then, on the ith step of the

transformation procedure, we will have the transformed ith block with new

354 V. Utkin et al.

state vector zi similar to (66). To carry out this, take the derivative of (67)along (51)–54)

zi = fi(xi, w) + Bi(xi)xi+1 (68)

where fi(xi, w) =∑i−1

j=1∂αi

∂xj[fj(xj) + Bj(xj)xi+1 + Dj(xj)w] + ∂αi

∂xi[fi(xi) +

Di(xi)w] + ∂αi

∂w s(w), Bi = BiBi, and rank Bi = ni. The fictitious controlinput xi+1 in (68) can be selected similar to (63) as

xi+1 = xci+1(xi, w)+B+

i (xi)(Kizi+Ei,1zi+1), xci+1 = −B+

i (xi)fi(xi, w)(69)

where zi+1 is an ni+1 × 1 vector of new variables, Ki is a desired matrix,Ei,1 =

[Ini

0], Ei,1 ∈ ni×ni+1 , with xc

i+1 calculated from the equationzi = 0 (68). Thus, equation (68) with (69) takes the same form as equation(66), namely zi = Kizi + Ei,1zi+1. For this step, we generalize assumptionB2 as follows:Bi) The elements of matrix Bi(xi) can be ordered such that

rankBi+1(xi) = ni+1, Bi+1(xi) =[Bi(xi)Ei,2

]

where Ei,2 =[0 Ini+1−ni

] ∈ (ni+1−ni)×ni+1 .Under this assumption, we can obtain from (69) the recursive transfor-

mation for zi+1 as

zi+1 = Bi+1(xi)xi+1 +[fi(xi, w) +Kiαi(xi, w)

0

]

: = αi+1(xi+1, w), i = 3, ..., r − 1 (70)

On the last step, after calculating the time derivative of zr = αr(x, w)(70), the original system (51)–54) is represented in the new coordinates z1,z2, ..., zr as

z1 = K1z1 + z2 (71)zi = Kizi + Ei,1zi+1, i = 2, ..., r − 1 (72)zr = fr(x, w) + Br(x)u+∆fr(x, w, v) (73)

where z = (z1, z2, ..., zr), fr(x,w) =∑r−1

j=1∂αi

∂xj [fj(xj) + Bj(xj)xi+1 +Dj(xj)w] + ∂αi

∂xr[fr(xr) + Dr(xr)w] + ∂αr

∂w s(w),∆fr(x, w, v) = ∂αi

∂xrFr(x)v,

Br = BrBr, and rank Br = nr.

Remark 2. Assumption (Bi) is not necessary. It is sufficient to use the ma-trix Ei,2 with constant parameters. It is clear that if this assumption is notsatisfied, then it will be possible to use for instance the matrix B⊥

i orthogonalto Bi, instead of Ei,2.

Note the equations (71)–(72) present uncontrolled dynamics while equa-tion (73) presents the controlled one.


6.3 Discontinuous Feedback Design

In the present setting, a natural choice for a switching function σ for system(71)–(73) is taking

σ = zr − cr(w) (74)

where

zr = Br(xr−1)xr+[fr−1(xr−1, w) +Kr−1αr−1(xr−1, w)

0

]:= αr(x, w)(75)

To generate sliding mode in (73), we use the fact that the term fr(x, w) inthe controlled block (73) is known, and matrix Br(x) has full rank, to achievea sliding mode on the manifold zr − cr(w) = 0 (74) by using the followingcombined control law:

u = uc(x, w)− krB−1r (x)sign(σ) (76)

where kr > 0, and uc(x, w) is the continuous control component chosen tocancel the known terms in (73), namely

uc(x, w) = −B−1r (x)

[fr(x, w)− ∂cr(w)

∂ws(w)

]. (77)

Note that in the absence of uncertainty, the control component uc(x, w) coin-cides with the equivalent control calculated as the solution of equation σ = 0.Substitution of (76)-(77) into (73) yields

σ = −krsign(σ) +∆fr(x, w, v) (78)

Then the controller (76)-(77) under the condition

kr > ‖∆fr(x, w, v)‖ (79)

guarantees a sliding mode on the surface zr − cr(w) = 0.

6.4 SMSP Solution Conditions

For the system constrained to the sliding manifold σ = 0, the system ((71)–(72) reduces to the following quasi-linear system of (n − nr) order

z1 = K1z1 + E1,1z2 (80)zi = Kizi + Ei,1zi+1, i = 2, ..., r − 2, (81)

zr−1 = Kr−1zr−1 + Er−1,1cr(w), (82)

with the output error

e = h(z1, ..., zr−1, c(w))− q(w) (83)

We can establish the following result.

356 V. Utkin et al.

Proposition 3. If1) Assumption H2 and H3 hold

2) There exist Ck (k ≥ 2) mappings πi(w), i = 1, ..., r − 1, c(w)withπi(0) = 0and c(0) = 0 which satisfy

Kiπi(w) + Ei,1πi+1 =∂πi(w)

∂ws(w), i = 1, ..., r − 2 (84)

Kr−1πr−1(w) + Er−1,1cr(w) =∂πr−1(w)

∂ws(w) (85)

h[π1(w), ..., πr−1(w), cr(w)]− q(w) = 0 (86)

Then the SMFSP is solvable.Proof. The systems (80)–(82) can be represented as

η = Aηη +Aww + ψ1(η, w) (87)w = Sw + ψ2(w) (88)

where η = (z1, ..., zr−1), and

Aη =

K1 In1 · · · 0 00 K2 · · · 0 0· · · · · · · · · · · · · · ·0 0 · · · Kr−2 Er−2,1

0 0 · · · 0 Kr−1

, Aw =

00· · ·0Cr

Cr = ∂c∂w (0) and S = ∂s

∂w (0) , with functions ψ1(w) and ψ2(w) vanishing at

the origin with its first derivatives. It is easy to see that there are matricesK1, ...,Kr−1 such that matrix Aη is Hurwitz. Therefore, the systems (87)and (88) under assumption H2 have a central manifold η = π(w) at (0, 0)with Ck mapping π(w), π(w) = (π1(w), ...πr−1(w))T , π(0) = 0) satisfyingthe condition (84)-(85). Since matrix Aη is Hurwitz, by a property of centermanifolds, η(t) → π(w(t)). So, by continuity, if condition (86) holds, then theoutput tracking error converges to zero.Remark. From the previous result we observe that, in the case of unmatcheduncertainty, the high gain technique can be applied to an exponentially stablemotion.

7 Application to the Pendubot

In this section we will apply the scheme proposed in section 6 to a model ofan underactuated system, the well-known Pendubot (Figure 1).

The dynamics of the Pendubot are described by the following equations


Fig. 1. Schematic diagram of the Pendubot

(D11(q2) D12(q2)D12(q2) D22(q2)

) (q1

q2

)+

(C1(q2, q1, q2)C2(q2, q1)

)+

(G1(q1, q2, )G2(q1, q2)

)=

(τ1

0

)

where q1 and q2 represent the generalized coordinates of the actuated and

the unactuated joints respectively. In our case, we have

D11 = m1l2c1 +m2(l21 + l2c2 + 2l1lc2 cos(q2)) + I1 + I2,

D12 = m2(l2c2 + 2l1lc2 cos(q2)) + I2

D22 = m2l2c2 + I2

C1 = −2m2l1lc2q1q2 sin(q2)− m2l1lc2q22 sin(q2),

C2 = m2l1lc2q22 sin(q2),

G1 = m1glc1 cos(q1) +m2gl1 cos(q1) +m2glc2 cos(q1 + q2)G2 = m2glc2 cos(q1 + q2)where the parameters values arem1 m2 l1 l2 lc1 lc2 g I1 I2

0.5289 0.3346 0.26987 0.38417 0.13494 0.19208 9.81 0.13863 0.016749

Choosing x = (x1, x2, x3, x4)T = (q2, q2, q1, q1, )T as state vector, u = τas control input, and y = q2 as output, the description of the system can begiven in the state space form as:

x = f(x) + b(x)u (89)y = h(x) (90)

wheref = (f1, f2, f3, f4, )T , b = (b1, b2, b3, b4, )T , h(x) = x1,

358 V. Utkin et al.

f1

f2

f3

f4

=

x2−D12

D11D22−D212

− c

x4D22

D11D22−D212

p1

;

b1

b2(x1)b3

b4(x1)

=

0−D12

D11D22−D212

0D22

D11D22−D212

with c = C2D22

− G2D22

, p1 = D12C2D22

− D12G2D22

− C1 − G1.

In order to apply the control scheme described in the previous sections tothe model of the Pendubot, suppose we are interested in tracking a referencesignal produced by the exosystem

w1 = αw2

w2 = −αw1

with α = 1 and w(0) =col(w1(0), w2(0)) , and e = x1 − w2.The corresponding Pfaffian system (43) in this case has the form

dx1 − a14(x1)dx4 = 0 (91)dx2 − a24(x1)dx4 = 0 (92)dx3 − a34(x1)dx4 = 0 (93)

with a14(x1) = b1b−14 (x1) = 0, a24(x1) = b2(x1)b−1

4 (x1) = −D12(x1)D−122 (x1)

and a34(x1) = b3b−14 (x1) = 0. Since a14(x1) = a24(x1) = 0 the equations (91)

and (93) have integrals x1 = C1 and x3 = C3. Therefore x1 can be consideredin (92) as a parameter and after integrating we have x2−a1(x1)x4 = C2 whereC1, C2 and C3 are the integration constants.

Choosing the transformation x′1 = x1, x′

2 = x2 − a1(x1)x4, x′3 = x3 and

x′4 = x4 system (89) is transformed into the regular form x = f ′(x) + b′(x)u

where

f ′1

f ′2

f ′3

f ′4

=

x′2 − D12D

−122 x′

4

−(c+G2)D−122 +D0x

′4

x′4

D22D11D22−D2

12p1

;

b′1b′2b′3b4

=

000

D22D11D22−D2

12

with D0 = −m2l1lc2 sin(x′1)

[x′

2 − D12D−122 x′

4

].

The system cannot be transformed to the NBC form, therefore we canapply the control scheme described for the regular form in section 5. Set

π = col(π1, π2, π3)z1 = x′

1 − π1(w)z2 = x′

2 − π2(w)z3 = x′

3 − π3(w)

and σ = x4 − σ1(z, w), σ1 = c1(w)− k1z1 − k2z2 − k3z3 where π1(w) = w2;k1, k2 and k3 are constant parameters, and π2(w), π3(w) and c1(w) are derived


from

−w1 = f ′1(π(w), c1(w))

∂π2(w)∂w

s(w) = f ′2(π(w), c1(w))

∂π3(w)∂w

s(w) = c1(w)

as a polynomial solution

π2(w) = −w1 + c1(w)π3(w) = −0.974377w2 + 0.0145446w2

1w2 − 0.00921993w32

c1(w) = 0.974374w1 − 0.0145446w31 − 0.05674899w2

1w2.

Finally, the control

u = ueq − kb−14 sign(σ),

ueq = −b−14

(f ′4 + k1f

′1 + k2f

′2 + k3f

′3 −

∂c1(w)∂w

), k = 10

is chosen to maintain sliding mode motion on σ = 0. To stabilize the linearapproximation of the sliding mode equation

z1 = 1.5961k1z1 + (1 + 1.5961k2)z2 + 1.5961k3z3

z2 = 21.6707z1 + 21.6707z3

z3 = −k1z1 − k2z2 − k3z3,

the set of parameters [k1, k2, k3] = [−34.8401,−7.4834,−40.6080] were used.In all the simulations, the initial conditions of the system were chosen

near the equilibrium point x1=π2 and x3 = 0 while variations in the nominal

values of mi and li were introduced.Figures 2 and 4 show the output responses of the closed loop systems

with nominal parameters, while Figures 3 and 5 show the behavior when a20% variation of the mass m2 is introduced. As we can see from Figure 5,the sliding mode based controller guarantees a small output tracking error,despite variation on the systems parameters.

8 Conclusion

The servomechanism problem for the variable structure system is introduced,and the solution conditions are derived for different classes of nonlinear sys-tems including systems in the so-called regular and NBC-forms. In particular,the combination of VSS and block control techniques allow the solution con-ditions to be obtained in a simpler way with respect to the classical setting. Inaddition, the sliding mode based controller achieves robustness with respectto uncertainties.

360 V. Utkin et al.

Fig. 2. Output signal response for the classical controller in absence of parameterperturbation.

Fig. 3. Output signal response for the classical controller under parameter varia-tions.

Fig. 4. Output signal response for the sliding mode controller in absence of param-eter perturbation.


Fig. 5. Output signal response for the sliding mode controller under parametervariations.

References

1. El-Chesawi, O.M.E., Zinober, A.S.I. and Billings, S.A., (1983), Analysis anddesign of variable structure systems using a geometric approach. InternationalJournal of Control Vol. 38, pp. 657-671.

2. Byrnes C.I., Priscoli, Delli F., Isidori A., and Kang W., (1997), Structurallystable output regulation of nonlinear systems, Automatica, Vol. 33, No.3, pp.369-385.

3. Goodall D.P. (1994): Lyapunov stabilization of a class of uncertain affine con-trol systems. -Lecture notes in control and Information Sciences 193. - VariableStructure and Lyapunov Control (A. Zinober, Ed.). Springer Verlag, New York.

4. Huang Jie and Ching-Fang Lin (1994). On a robust nonlinear servomechanismproblem. IEEE Trans. Aut. Control, Vol. 40, No.6, pp. 131-135.

5. Huang Jie, (1995). Asymptotic tracking and disturbance rejection in uncertainnonlinear systems. IEEE Trans. Aut. Control, Vol. 39, No.7, pp. 1510-1513.

6. Isidori A., Byrnes C.I. (1990), Output regulation of nonlinear systems, IEEETrans. Aut. Control, Vol 35, No.2, pp. 131-140.

7. Luk’yanov A.G. (1993), Optimal Nonlinear Block-Control Method. Proc. of the2rd European Control Conference, Groningen, pp. 1853-1855.

8. Loukianov, A.G. (1998), Nonlinear Block Control with Sliding Mode. Automa-tion and Remote Control, v. 59, No.7, pp. 916-933.

9. Luk’yanov A.G. and Utkin V.I. (1981), Methods for Reducing Dynamic Systemsto Regular Form, Automation and Remote Control, Vol. 42, No.4, (P.1), pp. 413-420. multivariable systems: a tutorial. -Proceedings IEEE 26, pp.1139-1144.

10. Utkin V.I. (1992), Sliding Modes in Control and Optimization Springer-Verlag,London.

9 Appendix

To obtain equation (44), at first, because the matrix B(x, t) does not havefull rank, we impose a constraint on the components of the control vector u,u = B+

1 (x, t)vo, such that

Bu = BB+1 vo =

[B12B

+1

In1

]vo (94)

362 V. Utkin et al.

where vo is a n1 × 1 input vector. It is easy to see, that , if ϕ1(x1, x12) = c isan integral of the Pfaffian system (43), the following identity

∂ϕ1

∂x1+

∂ϕ1

∂x12B2B

+1 ≡ 0 (95)

holds. Indeed, a differential dϕ1 calculated using the Pfaffian system (43), ofthe form

dϕ1 =∂ϕ1

∂xdx =

∂ϕ1

∂x1dx1 +

∂ϕ1

∂x12dx12 =

(∂ϕ1

∂x1+

∂ϕ1

∂x12B2B

+1

)dx1 (96)

is identically equal to zero. From this result, the identity (95) follows imme-diately.

Under assumption A12 there exist n1×1 vectors λ0(x, t) and ρ0(x, t) suchthat[

Ffm12

Fm1

]=

[B12B

+1

In1

]λ0 and

[Dm

12

Dm1

]=

[B12B

+1

In1

]ρ0

and next expression under condition (95)

∂ϕ1

∂x1Fm

1 +∂ϕ1

∂x12Fm

12 =(

∂ϕ1

∂x1+

∂ϕ1

∂x12B2B

+1

)λ0 = 0

∂ϕ1

∂x1Dm

1 w(t) +∂ϕ1

∂x12Dm

12w(t) =(

∂ϕ1

∂x1+

∂ϕ1

∂x12B2B

+1

)ρ0 = 0

hold. Differentiating the new variable x′2 = ϕ1(x1, x12) with respect to time

along trajectories of the system (39), gives

x′2 =

∂ϕ1

∂x1(f1 +B1u+ Fm

1 v + Fu1 v + (Dm

1 +Du1 )w

+∂ϕ1

∂x12(f12 +B12u+ (Fm

12 + Fu12)v + (Dm

12 +Du12)w +

∂ϕ1

∂t

=∂ϕ1

∂x1f1 +

∂ϕ1

∂x12f12 +

(∂ϕ1

∂x1+

∂ϕ1

∂x12B2B

+1

)v0 +

(∂ϕ1

∂x1+

∂ϕ1

∂x12B2B

+1

)λ0

+(

∂ϕ1

∂x1+

∂ϕ1

∂x12B2B

+1

)ρ0w + (Fu

1 + Fu12)v + (Du

1 +Du12)w

Defining

f ′2 =

(∂ϕ1

∂x1f1 +

∂ϕ1

∂x12f12 +

∂ϕ1

∂t

)x2=ϕ−1

1 (x1,x12,t)

D′2 = (Du

1 +Du12)x2=ϕ−1

1 (x1,x12,t)

F ′2 = (Fu

1 + Fu12)x2=ϕ−1

1 (x1,x12,t)

we obtain equation (44).


Variable Structure Systems Theory inComputational Intelligence

Mehmet Onder Efe1, Okyay Kaynak2, and Xinghuo Yu3

1 Carnegie Mellon University, Electrical and Computer Engineering DepartmentPittsburgh, PA 15213-3890, U.S.A.

2 Bogazici University, Electrical and Electronic Engineering Department Bebek,80815, Istanbul, Turkey

3 Faculty of Informatics and Communication, Central Queensland UniversityRockhampton QLD 4702, Australia

Abstract. Intelligence in the form of well-organized solutions to the ill-posedproblems has been the primary focus of many engineering applications. The ever-increasing developments in data fusion, sensor technology and high-speed micro-processors made the design in digital domain with high performance. A naturalconsequence of the progression during the last few decades is the emergence ofcomputationally intelligent systems. Neural networks and fuzzy inference systemsconstitute the core approaches of computational intelligence, whose methods haveextensively been used in the applications extending from image/pattern recogni-tion to identification and control of nonlinear systems. This chapter is devoted tothe analysis and design of learning strategies in the context of variable structuresystems. Several approaches are discussed in detail with special emphasis on thesliding mode control of nonlinear systems.

1 Introduction

Twentieth century has witnessed widespread innovations in all disciplines ofengineering sciences. Two snapshots from early 1900s and late 1990s differparticularly in terms of the active role of humans in performing complicatedtasks. The trend during the last century had the goal of implementing sys-tems having some degrees of intelligence to cope with the problem specificdifficulties that are likely to arise during the normal operation of the system.Today, it is apparent that the trend towards the development of autonomousmachinery will maintain its importance as the tasks and the systems aregetting more and more complicated. A natural consequence of the increasein the complexity of the task and physical hardware is to observe an ever-widening gap between the mathematical models and the physical reality towhich the models correspond. Having this picture in front of us, what now be-comes evident is the need for research towards the development of approacheshaving the capability of self-organization under the changing conditions ofthe task and the environment. Computational Intelligence (CI) is a frame-work offering various solutions to handle the complexity and the difficulties


of information-limited operating environments. The diversity in the solutionspace is a remarkable advantage that the designer utilizes either in the senseof algorithm-oriented manner or in the sense of architecture-oriented manner,hence, the result is an autonomous system exploiting these advantages.

Autonomy is one of the most important characteristics required from acomputationally intelligent system. A basic requirement in this context isthe ability to refresh and to refine the information content of the dynamics ofthe system. It therefore requires a careful consideration in the realm of engi-neering practice. From a systems and control engineering point of view, thedesigner is motivated by the time-varying nature of structural and environ-mental conditions to realize controllers that can accumulate the experienceand improve the mapping precision [1-2]. Methodologies imitating the infer-ence mechanism of the human brain are good in achieving the former andthose imitating the massively interconnected structure of the human brainare good in achieving the latter. In the literature, the linguistic aspects of in-telligence are discussed in the area Fuzzy Logic (FL) while the connectionistaspects are scrutinized in the area Neural Networks (NN). The integrationof these methodologies that exploit the strength of each collectively and syn-ergistically is a driving force to synthesize hybrid intelligent systems. Beingnot limited to what is mentioned, methods mimicking the process of evolu-tion, which are discussed under the title Genetic Algorithms (GA), and thoseadapted from artificial intelligence constitute other branches of CI and fallbeyond the focus of the approaches presented in this chapter.

NN are well known for their property of representing complex nonlinearmappings. Earlier works on the mapping properties of these architectureshave shown that NN are universal approximators [3-5]. The mathematicalpower of intelligence is commonly attributed to the neural systems because oftheir structurally complex interconnections and fault tolerant nature. Variousarchitectures of neural systems are studied in the literature. Feedforward andRecurrent Neural Networks (FNN, RNN) [6], Radial Basis Function NeuralNetworks (RBFNN) [1,6], dynamic neural networks [7], and Runge-Kuttaneural networks [8] constitute typical topologically different models.

FL is the most popular constituent of the CI area since fuzzy systems areable to represent human expertise in the form of IF antecedent THEN con-sequent statements. In this domain, the system behavior is modeled throughthe use of linguistic descriptions. Although the earliest work by Prof. LotfiZadeh on fuzzy systems [9] has not been paid as much attention as it deservedin the early 1960s, since then the methodology has become a well-developedframework. The typical architectures of Fuzzy Inference Systems (FIS) arethose introduced by Wang [10], Takagi and Sugeno [11] and Jang, Sun andMizutani [1]. In [10], a fuzzy system having Gaussian membership functions,product inference rule and weighted average defuzzifier is constructed andhas become the standard method in most applications. Takagi and Sugeno[11] change the defuzzification procedure where dynamic systems are used in

366 M.Ö. Efe, O. Kaynak, and X. Yu

the defuzzification stage. The potential advantage of the method is that un-der certain constraints, the stability of the system can be studied. Jang et al[1] propose an adaptive neuro-fuzzy inference system, in which polynomialsare used in the defuzzifier. This structure is commonly referred to as ANFISin the related literature.

When the applications of NN and FL are considered the process of learn-ing gains a vital importance. Although there is not a standard definition, theprocess of improving the future performance of the structures of CI by tuningthe parameters can be described as learning. The approaches existing in theliterature employ various techniques in achieving the desired parameter set(which is unknown), and require an iteratively evolving search mechanism.It should be noted that the most common technique that can be used in per-forming a suitable search operation in a multidimensional parameter space isbased on the use of an appropriately defined cost function. Alternatively, thesearch procedure can be implemented without using the derivative informa-tion; such as is done by the use of methods adapted from the evolutionarycomputation, e.g. GAs, or random search methods [1].

Error Backpropagation (EBP) technique [12] and Levenberg-Marquardt(LM) optimization technique [13] are the frequently used techniques used forparameter adaptation in CI. Both approaches are based on the utilizationof gradient information and necessitate the differentiability of the nonlinearactivation functions existing in the architecture with respect to the parame-ter to be updated, and frequently utilize some heuristics for improved real-ization performance. These typically concern the selection of learning rate,momentum coefficient, and adaptive learning rate strategies in EBP or step-size considerations in LM technique. However, the problem of convergenceor that of maintaining the bounded parameter evolution is an open problemassociated with these approaches. More explicitly, the learning strategy isnot protected against disturbances, which may excite the undesired internalmodes of EBP or LM approaches. The multidimensionality of the problem isanother difficulty in coming up with a thorough analysis distinguishing theuseful training information and disturbance-related excitation signals. Sincethe ultimate goal of the design is to meet the performance specifications,reducing the adverse effects of the disturbances requires that the adoptedlearning dynamics should be robustified. This steers the designer to seek formethods known in the conventional design framework. From this point ofview, a learning strategy based on Variable Structure Systems (VSS) theoryconstitutes a good candidate for eliminating the adverse effects of distur-bances.

VSS with sliding modes were first proposed in early 1950s [14-15]. How-ever, due to the implementation difficulties of high speed switching, it wasnot until 1970s that the approach received the attention it deserved. SlidingMode Control (SMC) technique nowadays enjoys a wide variety of applica-tion areas; such as in general motion control applications and robotics, in

367Variable Structure Systems Theory in Computational Intelligence

process control, in aerospace applications, and in power converters [16-19].The main reason for this popularity is the attractive properties that SMChave, such as good control performance for nonlinear systems, applicabilityto Multi-Input-Multi-Output (MIMO) systems, and well established designcriteria for discrete time systems. The most significant property of a slidingmode control system is its robustness. Loosely speaking, when a system is in asliding mode, it is insensitive to parameter changes or external disturbances.

From a systems and control theoretic point of view, the primary char-acteristic of variable structure control is that the feedback signal is discon-tinuous, switching on one or more manifolds in the state space. When thestate crosses each discontinuity surface, the structure of the feedback systemis altered. Under certain circumstances, all motions in the neighborhood ofthe manifold are directed towards the manifold and thus a sliding motionon a predefined subspace of the state-space is established in which the sys-tem state repeatedly crosses the switching surface [20]. This mode has usefulinvariance properties in the face of uncertainties in the plant model and there-fore is a good candidate for tracking control of uncertain nonlinear systems.The theory is well developed, especially for single-input systems in controllercanonical form.

The theory of VSS with sliding modes has been studied intensively bymany researchers. A recent comprehensive survey is given in [17] and vari-ous aspects of latest developments in VSS can be found in the chapters ofthis book. Motion control, especially in robotics, has been an area that hasattracted particular attention and numerous reports have appeared in theliterature [21-25]. One of the first experimental investigations that demon-strates the invariance property of a motion control system under a slidingmode is due to Kaynak et al [26].

In practical applications, a pure SMC approach suffers from the follow-ing disadvantages. Firstly, there is the problem of chattering, which is thehigh frequency oscillations of the controller output, brought about by thehigh speed (ideally at infinite frequency) switching necessary for the estab-lishment of a sliding mode. In practical implementations, chattering is highlyundesirable because it may excite unmodeled high frequency plant dynamicsand this can result in unforeseen instabilities.

Secondly, a SMC based feedback loop is extremely vulnerable to measure-ment noise since the control input depends tightly on the sign of a measuredquantity that is very close to zero [27]. Thirdly the SMC may employ unnec-essarily large control signals to overcome the parametric uncertainties. Lastbut not least, there exists appreciable difficulty in the calculation of whatis known as the equivalent control. A complete knowledge of the plant dy-namics is required for this purpose [28]. To alleviate these difficulties, severalmodifications to the original sliding control law have been proposed [29], themost popular being the boundary layer approach, which is, in essence, theapplication of a high gain feedback when the motion of the system reaches


ε-vicinity of a sliding manifold [22,28]. This approach is based on the ideaof the equivalence of the high gain systems and the systems with slidingmodes [30]. Another variation of the scheme is called provident control thatcombines variable structure control and variable structure adaptation andperforms hysteretic switching between the structures so as to avoid a slidingmode [31-32]. Both approaches are based on the calculation of the equivalentcontrol, requiring a good mathematical model of the plant.

The essence of the discussion presented in this chapter is to integrate VSStechnique and CI in an appropriate manner such that the difficulties of VSSapproach are alleviated by intelligence and the mathematical intractability ofintelligence is alleviated by VSS technique. Such a hybrid approach, particu-larly operating as the learning mechanism of CI architectures, is therefore agood candidate to represent the autonomous behavior of intelligent systemswith a robustified learning performance.

2 A Functional Overview of ComputationallyIntelligent Architectures

2.1 Adaptive Linear Elements (ADALINEs)

Being categorized as the basic operation in all architectures of CI, ADALINEperforms an inner product of two vectors, which is The output is a net sum inthe case of NNs, or the response of the system in the cases of RBFNN, SFS,ANFIS. The vectors of interest are the adjustable parameter vector and theexcitation input denoted by φ and u respectively. The input-output relationcan now be described as τ = φT u, where τ is the scalar output. Clearly, theapplications requiring multiple outputs τ would be a vector while φ wouldbe a matrix of appropriate dimensions.

2.2 Feedforward Neural Networks (FNNs)

FNNs constitute a class of NN structures in which the data flow is frominput to the output and no feedback connections are allowed. Because ofthe structural diversity of neural models, this discussion is devoted to thearchitecture and the mathematical representation of FNN structure, which isdiscussed from the point of control engineering. The architecture of a typicalFNN is illustrated in Figure 1, in which the neural network has three layersimplying the sufficiency for realizing any continuous mapping to a desireddegree of accuracy as long as the hidden layer contains sufficiently manyneurons [3-5]. The number of neurons in the hidden layer is a design variableand is mostly determined either by trial and error or by empirical results.

Functionally, o = ψh(Whu) and τ = ψ0(W0o), where ψh and ψ0 standfor the vectors of nonlinear activation functions for the hidden layer and theoutput layer respectively. Adaptation is carried out on the adjustable weights


u1

um

1

n

Input LayerVector Output: u

Hidden LayerVector Output: o

Output LayerVector Output: τ

τ

τ

Fig. 1. Structure of a FNN

contained in Wh and W0 matrices. In most applications of NNs, hyperbolictangent or sigmoid functions are used in ψh whereas the selection of ψ0 isgenerally a linear function of its argument, e.g. ψ0(x) = x. The standardapproach for tuning the parameters of FNNs is EBP or LM techniques [12-13].

Information contained in such a nonlinear map is distributed over its ar-chitectural constituents, i.e. neurons, such that a local failure in the structurecan be tolerated because of the parametric redundancy existing in the struc-ture, which is an analogue of the fault tolerance in biological systems. Moreexplicitly, the task can be redistributed upon death of neurons forming a localinfrastructure of a massive network.

2.3 Radial Basis Function Neural Networks (RBFNNs)

In the literature, RBFNNs are generally considered as a smooth transitionbetween FL and NNs. Structurally, a RBFNN is composed of receptive units(neurons) which act as the operators providing the information about theclass to which the input signal belongs. If the aggregation method, numberof receptive units in the hidden layer and the constant terms are equal tothose of a FIS, then there exists a functional equivalence between RBFNNand FIS [1]. Although the architectural view of a RBFNN is very similarto that of a FNN illustrated in Figure 1, the hidden neurons of a RBFNNpossess basis functions to characterize the partitions of the input space. Eachneuron in the hidden layer provides a degree of membership value for theinput pattern with respect to the basis vector of the receptive unit itself.The output layer is comprised of linear neurons. NN interpretation makesRBFNN useful in incorporating the mathematical tractability, especially inthe sense of propagating the error back through the network, while the FISinterpretation enables the incorporation of the expert knowledge into the


training procedure. The latter is of particular importance in assigning theinitial value of the network’s adjustable parameter vector to a vector thatis to be sought iteratively. Expectedly, this results in faster convergence inparameter space.

Mathematically, oi =∏m

j=1 ψij(uj) and a common choice for the hid-den layer activation function is the Gaussian curve described as ψij(u) =exp−(uj − cij)2/σ2

ij, where cij and σij stand for the center and the vari-ance of the ith neuron’s activation function qualifying the jth input variable.The output of the network is evaluated through the inner product of the ad-justable weight vector denoted by Φ and the vector of hidden layer outputs,i.e. τ = φT o, which is just as in the case of output evaluation in ADALINEs.Clearly the adjustable parameter set of the structure is composed of c,σ,φtriplet.

2.4 Standard Fuzzy Systems (SFSs)

Contrary to what is postulated in the realm of predicate logic, representationof knowledge by fuzzy quantities can provide extensive degrees of freedom ifthe task to be achieved can better be expressed in words than in numbers. Theconcept of fuzzy logic in this sense can be viewed as a generalization of binarylogic and refers to the manipulation of knowledge with sets, whose boundariesare unsharp [33]. Therefore the paradigm offers a possibility of designingintelligent controllers operating in an environment, in which the conditionsare inextricably intertwined, subject to uncertainties and impreciseness.

Understanding the information content of fuzzy logic systems is based onthe subjective judgements, intuitions and the experience of an expert. Fromthis point of view, a suitable way of expressing the expert knowledge is theuse of IF antecedent THEN consequent rules, which can easily evaluate thenecessary action to be executed for the current state of the system underinvestigation.

Structurally, a fuzzy controller is comprised of five building blocks, namely,fuzzification, inference engine, knowledge base, rule base, and defuzzification.Since the philosophy of the fuzzy models is based on the representation ofknowledge in fuzzy domain, the variables of interest are graded first. Thisgrading is performed through the evaluation of membership values of eachinput variable in terms of several class definitions. According to the defini-tion of a membership function, how the degree of confidence changes over thedomain of interest is characterized. This grading procedure is called fuzzifi-cation. In the knowledge base, the parameters of membership functions arestored. Rule base contains the cases likely to happen, and the correspond-ing actions for those cases through linguistic descriptions, i.e. the IF-THENstatements. The inference engine emulates the expert’s decision making ininterpreting and applying knowledge about how the best fulfillment of thetask is achieved. Finally, the defuzzifier converts the fuzzy decisions back ontothe crisp domain [34].


SFS architecture that has been proposed by Wang [35] uses algebraicproduct operator for the aggregation of the rule premises and bell-shapedmembership functions denoted by µ. The overall representation of SFS struc-ture is given in (1), in which R andm stand for the number of rules containedin the rule base and the number of inputs of the structure.

τ =R∑

i=1

fi

( ∏mj=1 µij (uj)∑R

i=1

∏mj=1 µij (uj)

)(1)

with ith rule as: IF u1 is U1i AND u2 is U2i AND . . . AND um is Umi THENfi = φi. In the IF part of this representation, the lowercase variables denotethe inputs and the uppercase variables stand for the fuzzy sets correspondingto the domain of each linguistic label. The THEN part is comprised of theprescribed decision in the form of a scalar number denoted by φi. Clearly, theadjustable parameters of the structure are comprised of the parameters of themembership functions together with the defuzzifier parameters φi. Anothercommon feature of the representation in (1) is the linearity of the output inthe defuzzifier parameters.

2.5 Adaptive Neuro-Fuzzy Inference Systems (ANFIS)

Adaptive neuro-fuzzy inference systems are synthesized by an appropriatelyintegrating the neural and fuzzy system interpretations. The resulting hybridcombination therefore inherits the numeric power of NN as well as the verbalpower of FL [1,36]. An ANFIS structure having m-inputs and single outputwith product inference rule and first order Sugeno model can be describedas in (1) with fi being described as in the rule consequent. The structuralview of such a system is illustrated in Figure 2, in which N stands for thenormalization operator seen as the last term of (1).

The rule structure for an ANFIS utilizing first order Sugeno model hasthe following representation: IF u1 is U1i AND u2 is U2i AND . . . AND um

is Umi THEN fi = φi,1u1+ . . .+φi,mum+φi,m+1. When the consequent partof the rule structure is compared with that of rules in SFS architecture, itis seen that the polynomial representation of the decision introduces higherparametric flexibility extending the realization capability. Being not confinedto what is discussed above, depending on the requirements of the problemin hand, the designer can choose higher order polynomials to improve therealization accuracy. When the issue of parameter tuning in ANFIS is con-sidered the well-known gradient approaches as well as the method of leastmean squares or VSS based approaches can easily be utilized.

ANFIS structure has been utilized with gradient based training strate-gies for identification of nonlinear systems [37] and with VSS based trainingstrategies for variable structure control of motion control systems. In [1], anin-depth discussion is given with numerous examples on the use of ANFISstructure.


Π Ñ

u1 um

Σ

u1

um

µ2m

u1 um

u1 um

τÑ

ÑΠ

Π

µ1m

µ11

µ21

µR1

µRm

Fig. 2. Structure of an ANFIS

3 VSS Based Parameter Tuning in Intelligent ControlSystems

The studies reporting the use of VSS for parameter tuning in CI by San-ner and Slotine [38], and Sira-Ramirez and Colina-Morles [39] have been thestimulants, which proved that the robustness feature of VSS could be ex-ploited in the training of the architectures of CI. These studies pioneereda vast majority of researchers working on VSS and CI. Sanner and Slotineconsidered the training of GRBFNN which has certain degrees of analyticaltractability in explaining the stability issues, and Sira-Ramirez et al haveshown the use of ADALINEs with a VSS based learning strategy. As an illus-trative example, the inverse dynamics identification of a Kapitsa pendulumhas been demonstrated together with a thorough analysis towards the han-dling of disturbances. Hsu and Real [40-41] demonstrate the use of VSS withGaussian NNs, Yu et al [42] introduces the dynamic uncertainty adaptationof what is proposed in [39], and demonstrate the performance of the schemeon the Kapitsa pendulum. Parma et al [43] use the VSS technique in pa-rameter tuning process of multilayer perceptron. Latest studies towards theintegration of VSS and CI have shown that the tuning can be implementedin dynamic weight filter neurons [44], in parameters of a controller [45]. Adifferent viewpoint towards this integration is due to Efe et al [46-47], whichhas the goal of reducing the adverse effects of noise driven parameter tuningactivity in gradient techniques. The key idea in these works is to mix thetwo training signals in a weighted average sense. A good deal of review isprovided in the recent survey of Kaynak et al [48]. The survey illustrates how


VSS can be used for training in CI as well as how CI can be utilized for thetuning of parameters in conventional VSS.

In what follows, the use of VSS approach for intelligent control of non-linear systems is presented together with the analytical details wherever re-quired. The emphasis is mainly on the works presented in [44-45] with theauthors’ latest research outcomes towards the direction of control engineer-ing.

3.1 Control System Structure

Consider the feedback loop illustrated in Figure 3, in which a subscript ddenotes the desired value of the relevant quantity. Furthermore, it is shownin the figure that if a supervisor provides the desired controller outputs, onemight evaluate the error on the control signal denoted by sc.

INTELLIGENTCONTROLLER PLANT

Σ

Σ

sc τd

θd

+

_

+_ τ

θ

Fig. 3. Block diagram of the control system

The plant shown in Figure 3 is assumed to have the structure describedin (2), in which θ and τ are (r1 + r2 + . . .+ rn)×1−dimensional state vectorand n × 1−dimensional input vector. The system of (2) with these vectorscan be restated as θ = fp (θ) +Dτ .

θ(ri)i = fpi

(θ) +n∑

j=1

dijτj i = 1, 2, . . . , n (2)

The design problem is to enforce the behavior of the system towards thedesired response, which is known but the control signal (τ d) resulting inwhich is unavailable. Therefore, the solution to this problem is a search to-wards the synthesis of such a signal iteratively by an intelligent controller.Assuming that the intelligent controller in Figure 3 is composed of n individ-ual controllers, the ith one of which is to construct the ith component of inputvector τ , the jth entry of the error vector driving this sub-controller can be


given as e(j)i = θ(j)i − θ(j)di . Apparently, this component is the jth derivative

of the relevant state component.

3.2 Conventional VSS Design - An Overview

Consider the vector of sliding surfaces for the system in (2): sp(e) = Ge =G(θ − θd). The widespread selection of the matrix G is such that the ith

sliding surface function has the form

spi(ei) =

(d

dt+ λi

)ri−1

ei (3)

in which, λi is a strictly positive constant. Let Vp be a candidate Lyapunovfunction given as

Vp (sp) =12sT

p sp (4)

If the prescribed control signal satisfies Vp (sp) = −sTpΞ sgn(sp), the neg-

ative definiteness of the time derivative of the Lyapunov function in (4) isensured. In above, Ξ is a positive definite diagonal matrix of dimensionn × n. More explicitly, sT

p sp = −sTpΞ sgn(sp) must hold true to drive the

error vector towards the sliding hypersurface. On the other hand, the use ofsp = −Gθd +G

(fp (θ) +Dτ

)leads to the following control signal:

τ = − (GD)−1(Gfp (θ)−Gθd

)− (GD)−1

Ξ sgn(sp) (5)

in which, the first term is the equivalent control term and the second term isthe corrective control term. For the existence of the mentioned components,the matrixGD must not be rank deficient. In the literature, equivalent controlis considered as the low frequency (average) component of the control signal.Because of the discontinuity on the sliding surface, the corrective term bringsa high rate component [20,25]. If e(0) = 0, the tracking problem can beconsidered as keeping e on the sliding surface, however, for nonzero initialconditions, the strategy must enforce the state trajectories towards the slidingsurface, which is ensured by the negative definiteness of the time derivativeof the Lyapunov function as in (4). For the case of nonzero initial conditions,the phase until the error vector hits the sliding surface is called the reachingmode, the dynamic characteristics of the system during which is determinedby the control strategy adopted. Application of the control input formulatedin (5) imposes the dynamics described as sp = −Ξ sgn(sp), which clearlyenforce the error vector towards the sliding surface. Once the sliding surfaceis reached, the value of (3) becomes zero; and this enforces the error vectorto move towards the origin.

Aside from the practical difficulties of conventional VSS schemes, thecontrol signal in (5) is applicable if a nominal representation of the systemunder control is available. In the next subsection, a method for obtaining theerror on the control signal is presented for unknown systems of structure (2).


3.3 Calculation of the Control Error

Remark 3.1: The VSS task is achievable if the dynamics of the system in(2) is totally known or if the nominal system is known with the bounds ofthe uncertainties. It must be noted that to satisfy the matching conditions,the disturbances and uncertainties are always assumed to enter the systemthrough the control channels [17]. When the conventional VSS strategy isapplied to the system of (2), we call the resulting behavior as the target VSSand the input vector leading to it as the target control sequence (τ ), which isdescribed in (5). If the functional form of the vector function fp is not known,it should be obvious that the target control sequence cannot be constructedby following the traditional VSS design approaches.

Definition 3.2: Given an uncertain plant, which has the structure describedas in (2), and a command trajectory vector θd(t) for t ≥ 0, the input sequencesatisfying the following vector differential equation is defined to be the ide-alized control sequence denoted by τ d, and the vector differential equationitself is defined to be the reference SMC model.

θd = fp (θd) +Dτ d (6)

Mathematically, the existence of such a model and the sequence meansthat the system of (2) perfectly follows the command trajectory vector ifboth the idealized control sequence is known and the initial conditions areset as θ(t = 0) = θd(t = 0), more explicitly e(t) ≡ 0 for t ≥ 0. Undoubtedly,such an idealized control sequence will not be a norm-bounded signal whenthere are step-like changes in the vector of command trajectories or when theinitial errors are nonzero. It is therefore that the reference SMC model is anabstraction due to the limitations of the physical reality, but the concept ofidealized control sequence should be viewed as the synthesis of the commandsignal θd from the time solution of the differential equation set in (6).

Fact 3.3: Based on the Lyapunov stability results of the previous subsection,if the target control sequence formulated in (5) were applied to the systemof (2), the idealized control sequence would be the steady state solution ofthe control signal, i.e. limt→∞ τ = τ d. However, under the assumption of theachievability of the VSS task, the difficulty here is again the unavailabilityof the functional form of the vector function fp. Therefore, the aim in thissubsection is to discover an equivalent form of the discrepancy between thecontrol applied to the system and its target value by utilizing the idealizedcontrol viewpoint. This discrepancy measure is denoted by sc = τ − τ d andis of n× 1 dimension.

If the target control sequence of (5) is rewritten by using (6), one gets

τ = − (GD)−1(Gfp (θ)−G

(fp (θd) +Dτ d

)+Ξ sgn(sp)

)


= − (GD)−1(Gfp (θ)−Gfp (θd) +Ξ sgn(sp)

)+ τ d (7)

= − (GD)−1(G∆fp (θ) +Ξ sgn(sp)

)+ τ d

The target control sequence becomes identical to the idealized control se-quence, i.e. τ ≡ τ d, as long asG∆fp (θ)+Ξ sgn(sp) = 0 holds true. However,this condition is of no practical importance as we do not have the analyticform of the vector function fp. Therefore, one should consider this equalityas an equality to be enforced instead of an equality that holds true all thetime, because its implication is sc = 0 and is the aim of the design. It is ob-vious that to enforce this to hold true will let us synthesize the target controlsequence, which will ultimately converge to the idealized control sequence bythe adaptation algorithm yet to be discussed. Consider the time derivative ofthe vector of sliding surfaces

sp (e) = Ge

= G(θ − θd)

= G(fp (θ) +Dτ − fp (θ)d −Dτ d

)(8)

= G(∆fp +D (τ − τ d)

)

= G(∆fp +Dsc

)

Utilizing G∆fp+Ξ sgn(sp) = 0 in (8) and solving for sc yields the followingrelation:

sc = (GD)−1(sp +Ξ sgn(sp)

)= τ − τ d (9)

Remark 3.4: The reader must here notice that the application of τ d to thesystem of (2) with zero initial errors will lead to e(t) ≡ 0 for ∀t ≥ 0, onthe other hand, the application of τ to the system of (2) will lead to sp = 0for ∀t ≥ th, where th is the hitting time, and the origin will be reachedaccording to the dynamics of the sliding surface. Therefore, the adoption of(9) as the equivalent measure of the control error loosens e(t) ≡ 0 for ∀t ≥ 0requirement and introduces all trajectories in the error space to tend to thesliding hypersurface, i.e.G∆fp+Ξ sgn(sp) = 0 is enforced. Consequently, thetendency of the control scheme will be to generate the target VSS sequenceof (5) without requiring the analytical details of the plant.

Now consider the ordinary feedback control loop illustrated in Figure 3,and define the following Lyapunov function, which is a measure of how wellthe controller performs:

Vc (sc) =12sT

c sc (10)


Remark 3.5: An adaptation algorithm ensuring Vc (sc) < 0 when sc = 0enforces G∆fp + Ξ sgn(sp) = 0 and creates the predefined sliding regimeafter a reaching mode lasting until the hitting time denoted by th, beyondwhich sc = 0 as the system is in the sliding regime. If Vc (sc) < 0 whensc = 0, then limt→th

Vc = 0 ⇐⇒ limt→th‖ sc ‖= 0 ⇐⇒ limt→th

‖sp + Ξ sgn(sp) ‖= 0. Note that the meaning of sc = 0 is now equivalent tosp = 0 by Remark 3.4, therefore the limits above are evaluated as t → th.

3.4 Parameter Tuning based on a Single-Term LyapunovFunction

If the architectures introduced in the second section are utilized for the pur-pose of control, without loss of generality, the output of the ith controllercan be restated as τi = φT

i Ωi, where Ωi is the vector of signals exciting theadjustable parameters denoted by φi. Therefore the algorithm discussed hereis applicable to ADALINE, GRBFNN, SFS and ANFIS architectures. Fur-thermore, the Lyapunov function in (10) constitutes the basis of the design.

In order not to be in conflict with the physical reality, the designer mustimpose ‖ φi ‖≤ Bφi

, ‖ Ωi ‖≤ BΩi, ‖ Ωi ‖≤ BΩi

, and ‖ τid ‖≤ Bτidthe

truth of which state that the adjustable parameters of the controller, thetime derivative of the signal exciting the adjustable parameter set and thetime derivative of the idealized output of the controller remain bounded.Note that in Definition 3.2, we stated that there may not be a finite Bτid

∈ even in some realistic situations like nonzero initial errors, however, thepractical meaning of imposing ‖ τid ‖≤ Bτid

will lead us to the constructionof an approximation of the idealized control sequence and the requirement ofe(t) ≡ 0 for ∀t ≥ 0 must therefore be loosened.

Theorem 3.6: For the ith subsystem of the system described in (2), adopt-ing the controller of structure τi = φT

i Ωi, the adaptation of the controllerparameters as described in (11) enforces the value of the ith component ofcontrol discrepancy vector (sci

) to zero.

φi = − Ωi

ΩTi Ωi

ki sgn(sci) (11)

where, ki is a sufficiently large positive constant satisfying ki > BφiBΩi

+Bτid

. The adaptation mechanism in (11) drives an arbitrary initial value ofsci to zero in finite time denoted by thi satisfying the inequality in (12).

thi ≤ | sci(0) |

ki −(Bφi

BΩi+Bτid

) (12)

Proof: See Sira-Ramirez et al [39] and Efe et al [45].

An important feature of this approach is the fact that the controller pa-rameters evolve bounded as assumed initially. The details of the boundedparametric evolution analysis can be found in [42,45].


3.5 Parameter Tuning based on a Two-Term Lyapunov Function

Similar to what is initially designated in the previous subsection, the outputof the ith controller is described as τi = φT

i Ωi. In addition to the statedboundedness conditions the truth of ‖ Ωi ‖≤ BΩi

is imposed. Consider theLyapunov function given in (13), in which µ and ρ are the weights to beselected by the designer.

V = µVci+ ρ

12

∥∥∥∥∂Vci

∂φi

∥∥∥∥2

with Vci=

12s2ci

(13)

Theorem 3.7: If the adaptation strategy for the adjustable parameters ofthe ith controller is chosen as

φi = −ki

(µI + ρ

∂2Vci

∂φi∂φTi

)−1

sgn(∂Vci

∂φi

)(14)

with ki is a sufficiently large constant satisfying ki > (µBφi+ ρBΩi

)BΩi,

then the negative definiteness of the time derivative of the Lyapunov functionin (13) is ensured.

Proof: See Efe [49].

3.6 A Generalization of EBP and LM Techniques in the Contextof VSS

A recent contribution towards the generalization of EBP and LM techniquesis due to Yu et al [50]. The approach postulated is applicable to all architec-tures discussed in the second section and is based on the Lyapunov functiongiven in (15).

V (Ji,φi) = µJi + ρ12

∥∥∥∥ ∂Ji

∂φi

∥∥∥∥2

(15)

where Ji = γ−1∫ t

t−γsci

(σ) dσ with γ being the length of a time window toevaluate the training efficiency [51-52].

Theorem 3.8: For a computationally intelligent structure whose input-output relationship is τi(t) = (φi(t),ui(t)), if

(a) ∂Ji

∂t < 0 and

(b) The parameter adaptation rule is


φi =

−(µI + ρ ∂2Ji

∂φi∂φTi

)−1(

∂Ji∂φT∥∥ ∂Ji∂φi

∥∥2

)∗

(µ∂Ji

∂t + ρ ∂Ji

∂φi

∂2Ji

∂t∂φTi

+ ζ∥∥∥ ∂Ji

∂φi

∥∥∥2

+ ηJi

)if

∥∥∥ ∂Ji

∂φi

∥∥∥ = 0

0 otherwise

(16)

Then ∂Ji

∂φitends to zero asymptotically.

Proof: See Yu et al [50].

The formulation of Ji is particularly useful for on line training and contin-uous time learning. However, for discrete data, since the evaluation of errorscan only be done at discrete instants of time, Ji at time tk can be defined asJi(t = tk) = limγ→0 γ

−1∫ t

t−γsci

(σ) dσ = sci(tk). The conventional gradient

descent learning algorithm can be now obtained by setting ρ = 0 and η = 0.Since ∂2Ji(t=tk)

∂t∂φTi

= 0 one obtains the law in (17), whose learning rate in the

conventional sense is η−1ζ.

φ = −η−1ζ∂Ji(t = tk)∂φT

(17)

The Gauss-Newton algorithm can be obtained by setting µ = 0 and η = 0.Since ∂Ji(t=tk)

∂t = 0 and ∂2Ji(t=tk)

∂t∂φTi

= 0, from (16) one gets the law in (18).

φi =(σ∂2Ji(t = tk)∂φi∂φT

i

)−1 (ζ∂Ji(t = tk)∂φT

i

)

= −σ−1ζ

(∂2Ji(t = tk)∂φi∂φT

i

)−1 (∂Ji(t = tk)∂φT

i

)(18)

Similarly, the LM algorithm can be easily obtained by setting η = 0. Since∂Ji(t=tk)

∂t = 0 and ∂2Ji(t=tk)

∂t∂φTi

= 0, from (16) the law in (19) is obtained.

φi = −(µI + σ

∂2Ji(t = tk)∂φi∂φT

i

)−1 (ζ∂Ji(t = tk)∂φT

i

)(19)

3.7 Practical Issues

The analysis and the design approach presented so far have tried to illumi-nate the VSS based training problem from a theoretical perspective. In thissubsection, we discuss several issues related to the practical applications ofthe discussed methodologies.


Chattering Since the control decision during the sliding mode is tightlydependent to the sign of a measured quantity being noisy and very close tozero, the decision along the sliding manifold exhibits sensitivity to noise onthe observations. Among many alternatives available [17,28,53], a commonapproach to eliminate the chattering is to smooth the sign function, whichcorresponds to introduce a boundary layer [28]. A widespread choice is thefollowing approximation for the sgn(.) function.

sgn(x) ∼= x

|x|+ δ (20)

where δ determines the sharpness around the origin. Since the function in(20) is not discontinuous at the origin, the decision mechanism softly switchesinside the boundary layer.

Actuation Speed Another important issue is the actuation speed of thesystem under control, i.e. the ability to respond to what is imposed timely.Since the details concerning the dynamic model of the plant under controlare assumed to be unavailable, what causes a difficulty from a practical pointof view is the selection of the matrix Ξ, which characterizes the behaviorduring the reaching mode. The values of this quantity can only be set bytrial-and-error due to the lack of system-specific details.

Obtaining the Equivalent Error from the Observed Data Lastly inthis subsection, we focus on the construction of the sc of (9), which requiresthe differentiation of sp. A suitable approach is to filter the measured valuesof sp and differentiate afterwards. Denote S as the Laplace variable, and usethe linear dynamic system given as

H(S) =αS

Q(S)(21)

where Q(0) = α > 0 and Realroots(Q(S)) < 0. The order of the de-nominator polynomial and the locations of the roots are left to the designer,because these issues require several trials to refine the selections and are sub-ject to the application together with its operating environment. It shouldbe noted that the cost of the information loss by using such a filter, whoseinput is sp and output is an estimate of sp, is a matter of how robust thedevised control algorithm is. More explicitly, the separation of the noise andthe actual value of sp leads to a corruption on sp, and when differentiatedafterwards, some valuable information is lost together with the eliminationof the noise component. Here it is assumed that the mentioned loss causesan uncertainty, which enters the system through the control channels, andwhich is particularly effective during the sliding mode; and this uncertaintycan be alleviated if it falls within the limits allowing the maintenance of theinvariance during the sliding mode [17].


Computational Burden One of the factors qualifying the physical imple-mentability of control schemes is the number of computations to be performedby the controller. In this part, a discussion of the computational burden ofthe tuning mechanisms is presented. It should be noted that the structureof the controller adopted strictly influences the required number of float-ing point operations (flops) between the two consecutive sampling instants.Therefore, the discussion given here focuses on the ADALINE controller, as itconstitutes a basis of all structures. If an ANFIS structure is to be used, thedesigner must consider the extra calculations to generate the vector signalexciting the adjustable parameter set of the defuzzifier.

Another point to clarify is the computational complexity due to the ap-proach postulated in Theorem 3.8, whose practical applications generallysubject to the following: the cost function Ji is evaluated at the discrete in-stants of time and it does not depend explicitly on time, i.e. ∂Ji(t=tk)

∂t = 0

and ∂2Ji(t=tk)

∂t∂φTi

= 0,.Figure 4 illustrates a bar graph composed of triplets. The leftmost com-

ponent represents the flops required to evaluate the ADALINE output and toadjust its parameters once by utilizing the method discussed in the subsection3.4. The middle and the rightmost components stand for the required numberof flops for the methods presented in subsections 3.5 and 3.6 respectively. It isclear from the figure that the complexity due to the first approach is consid-erably smaller than the other two as the order of the subsystem under controlincreases. This fact is primarily because of the matrix inversion to be per-formed at each step. However, the set of criteria qualifying the performanceof an intelligent control system is strictly dependent upon the applicationspecific details, which does not give a clue in choosing a tuning mechanism.Therefore, the designer is encouraged to try the alternatives in discoveringthe one performing the best.

3.8 Summary

What we have discussed so far have illuminated the design considerations atmicroscopic levels forming the whole picture. When implementing the controlsystem with a VSS based tuning mechanism updating the parameters of anintelligent controller, one has to remember that the plant is in an ordinaryfeedback loop as illustrated in Figure 3. Having decided on the controllerstructure, the error vector is processed until the control to be applied isobtained. Since the desired control inputs are unavailable, using the errormeasure given in (9), the similarity between the applied control and thetarget control sequence is qualified, then the parameter tuning is performedaccording to the chosen tuning strategy.

A particular difference in applying the ADALINE structure as the con-troller with (11) and (14) is that the controller input vector is formed byaugmenting the error vector, which is of dimension ri × 1, with a constant


1 2 3 4 5 6 7 80

500

1000

1500

2000

2500

3000

Order of the subsystem under control (ri )

Req

uire

d nu

mbe

r of

flop

s1st Method

2nd Method

3rd Method

Fig. 4. Computational burden of the discussed schemes

bias of value unity yielding a (ri + 1)× 1-dimensional excitation to the con-troller. The reason for such an augmentation is twofold:

i) If the denominator of (11) were considered, without such an augmentationthe derivative would tend to infinity as the error vector moves towards theorigin. However, having such a tendency in the adjustable controller param-eters cannot result in convergence. When (14) is considered from the samepoint of view, together with the open form of matrix inversion, one sees thata convergent behavior enforces the tuning mechanism to behave like gradientdescent. Although gradient descent can appropriately be used for controllertraining purposes, the structural simplicity of ADALINE will not allow theobservation of a convergent behavior. This particular structure corresponds tolinear time varying state feedback, which is well developed especially for sys-tems whose dynamic representations are known totally or partly with knownuncertainty bounds.

ii) When the sliding mode starts, the error vector rapidly converges to ori-gin and the system starts tracking the desired trajectory precisely. However,since the magnitudes of the entries of the error vector are very close to zero,the corresponding controller parameters do not receive sufficient excitation tomaintain the synthesis of target control sequence. In implementing RBFNN,


SFS and ANFIS architectures, the designer will not need such an augmenta-tion since the parameter vector is persistently excited by the hidden neuronoutputs of RBFNN or rule outputs of FL based structures.

A last remark here is on the applicability of FNN structure, to which solelythe method in subsection 3.6 is applicable among what we have discussed.

4 An Illustrative Example

This section demonstrates the performance of the algorithm discussed insubsection 3.4 for a third order system studied previously by Roy et al [54]and Yilmaz et al [55]. The dynamic equation describing the system is givenin (22).

θ(3) = −0.5θ − 0.5θ3 − 0.5θ∣∣∣θ

∣∣∣ +(1 + 0.1 sin

(πt

3

))τ + κ1(t) + κ2(t) +

(−0.05 + 0.25 sin (5πt)) θ + (−0.03 + 0.3 cos (5πt)) θ3 + (22)

(−0.05 + 0.25 sin (7πt)) θ∣∣∣θ

∣∣∣

where κ1(t) = 0.2 sin(4πt) is the disturbance used in [54-55], and κ2(t) isthe zero mean Gaussian noise corrupting the state information to be usedby the controller additively. The work presented by Roy et al assume thatthe nominal system dynamics is known and the uncertain part is comprisedof what we give as the last three terms in (22). The primary difference be-tween what has been discussed so far and what is assumed in [54] should bestressed as the approaches we discuss only assume the achievability of theVSS task, hence the uncertainties are represented in the system dynamics,whose form is known but the details are not. As the controller, a three inputsingle output ANFIS structure is used and the tuning is performed only onthe defuzzifier parameters, which are initially set to zero. The rule base has27 rules quantifying the relevant input variable as Negative, Zero or Positive.Once the rule outputs are evaluated, the crisp decision of the controller iscomputed as described in (1).

Parallel to [54], the reference state trajectory, which is described as θd =0.5 cos(πt/5) is used in the simulations. Initially, the states of the system havethe following values, θ(0) = 1, θ(0) = 1 and θ(0) = 1. One important notehere should be on the selection of λ. The value is taken as 5 in [54]; howeverwe use λ = 1, because the behavior with this value results in a better systemresponse. Figure 5 illustrates the trajectory followed in the phase space. Theerror vector hits the sliding surface several times and starts moving on it asenforced by the algorithm.


0

0.2

0.4

0.6

-0.200.20.40.60.81

-6

-4

-2

0

2

de(t)/dte(t)

d2e(

t)/dt

2

t ≈ 0.25sec

Origin of the phase space

Fig. 5. Behavior in the phase space

5 Computational Intelligence in Variable StructureControl

What we have focused on so far mainly contemplates the use of VSS theoryfor parameter adaptation in CI. However, the integration of VSS techniquewith the architectural and algorithmic methods of CI can also be utilized in

• Chattering elimination through filtering [48,56]• Design of the parameters of a conventional sliding mode controller [48,57-

58]• Modeling of the uncertainties [48,59-61]• Generating a complementary control action [48,62-63]• Generating the equivalent control and corrective control actions sepa-

rately [48,64].

The use of CI in VSS may be a remedy in the situations where the avail-able knowledge is insufficient to produce a safe control action. The selectionof the uncertainty bound in this respect constitutes an apparent example.As the value of the uncertainty bound increases, the produced control actionis more likely to have high frequency components having high magnitude,which arise through the sign measurement during the sliding mode. In sucha situation, CI supported schemes can offer smoothed control signals with a


reasonable uncertainty bound selection. Furthermore, the conventional frame-work may underestimate the actuation speed of the system under control andmay lead to unnecessarily large control inputs. In these cases, the tuning ofVSS parameters, e.g. the slope of the sliding line, can be designed using themethods of CI. Being not limited to these, the methods of CI can be used asauxiliary subsystems for improving the control signal, i.e. a complementarycontrol signal is produced so that the undesired effects of conventional slidingmode controller can be reduced. Last but not the least, the components of thecontrol signal driving the system behavior to a predefined sliding regime canseparately be realized by a learning system. This can eventually result in acomprehensible way of formulating the equivalent control and the correctivecontrol.

6 Conclusion

This study discusses the design of a VSS theory based training strategies forCI, when the traditional gradient based training approaches are utilized forwhich, some handicaps arise due to the imperfect modeling, noisy observa-tions or time varying parameters. If the effects of these factors are trans-formed to the cost hypersurface, whose dimensionality is determined by theadjustable design parameters, it becomes evident that the surface may havedirections along which the sensitivity derivatives assume large values. In thesecases, gradient based optimization procedures tend to evaluate large paramet-ric displacements, which can eventually lead to a locally divergent behavior.In control engineering practice, such a behavior constitutes a potential dan-ger from a safety point of view. The approaches presented in this work takecare of the mentioned difficulties. Since the VSS theory is well known with itsrobustness property, a training strategy equipped with which retains a highdegree of robustness against disturbances and uncertainties. When these ap-proaches are considered for the training of intelligent controllers, under theassumption that the VSS task is achievable, the task is fulfilled without know-ing the analytic details describing the plant dynamics. In order to corroboratethe performance claims, tracking control of a third order nonlinear system ispresented. The behavior in the phase space clearly demonstrates the superiorperformance despite the unavailability of system-specific details.

References

1. Jang J.-S.R., Sun C.-T., Mizutani E. (1997) Neuro-Fuzzy and Soft Computing,PTR Prentice-Hall

2. Berenji H. (1992) Fuzzy and Neural Control, In: P. J. Antsaklis, and K. M.Passino (Eds.), An Introduction to Intelligent and Autonomous Control, 215-236, Kluwer Academic Publishers, The Netherlands

3. Hornik K. (1989) Multilayer Feedforward Networks are Universal Approxima-tors, Neural Networks, 2, 359-366


4. Funahashi, K. (1989) On the Approximate Realization of Continuous Mappingsby Neural Networks, Neural Networks, 2, 183-192

5. Cybenko G. (1989) Approximation by Superpositions of a Sigmoidal Function,Mathematics of Control, Signals, and Systems, 2, 303-314

6. Haykin S. (1994) Neural Networks, Macmillan College Printing Company, NewJersey

7. Gupta M.M., Rao D.H. (1993) Dynamical Neural Units with Applications tothe Control of Unknown Nonlinear Systems, Journal of Intelligent and FuzzySystems, 1, 1, 73-92

8. Wang Y-J., Lin C-T. (1998) Runge-Kutta Neural Network for Identification ofDynamical Systems in High Accuracy, IEEE Transactions on Neural Networks,9, 2, 294-307, March

9. Zadeh L.A. (1965) Fuzzy Sets, Information and Control, 8, 338-35310. Wang L. (1997) A Course in Fuzzy Systems and Control, PTR Prentice-Hall11. Takagi T., and Sugeno M. (1985) Fuzzy Identification of Systems and Its Ap-

plications to Modeling and Control, IEEE Transactions on Systems, Man, andCybernetics, 15, 1, 116-132, January

12. Rumelhart D.E., Hinton G.E., and Williams R.J. (1986) Learning Internal Rep-resentations by Error Propagation, in D. E. Rumelhart and J. L. McClelland,(Eds.), Parallel Distributed Processing: Explorations in the Microstructure ofCognition, 1, 318-362, MIT Press, Cambridge, M.A.

13. Hagan M. T., M. B. Menhaj (1994) Training Feedforward Networks with theMarquardt Algorithm, IEEE Transactions on Neural Networks, 5, 6, 989-993,November

14. Emelyanov S. V. (1959) Control of First Order Delay Systems by means ofan Astatic Controller and Nonlinear Correction, Automat. Remote Contr., 8,983-991

15. Utkin V. I. (1977) Variable Structure Systems with Sliding Modes, IEEE Trans-actions Automatic Control, 22, 212-222

16. Young K. D., Utkin V. I., Ozguner U. (1999) A Control Engineer’s Guide toSliding Mode Control, IEEE Transactions on Control Systems Technology, 7,3, 328-342, May

17. Hung J.Y., Gao W., Hung J. C. (1993) Variable Structure Control: A survey,IEEE Transactions on Industrial Electronics, 40, 1, 1-9, February

18. Zinober A.S.I. (1994) (Ed), Variable Structure and Lyapunov Control, Springer-Verlag

19. Young K.K. (1993) (Ed), Variable Structure Control for Robotics andAerospace Systems, Elsevier-Science

20. Utkin V.I. (1992) Sliding Modes in Control Optimization, Springer Verlag, NewYork

21. Young K.D. (1978) Controller Design for a Manipulator Using Theory of Vari-able Structure Systems, IEEE Transactions on Systems, Man, and Cybernetics,SMC-8, 210-218

22. Slotine J.J.E., Sastry S.S. (1983) Tracking Control of Nonlinear Systems UsingSliding Surfaces with Application to Robot Manipulators, International Journalof Control, 38, 465-492

23. Hashimoto H., Maruyama K., Harashima F. (1987) A Microprocessor BasedRobot Manipulator Control with Sliding Mode, IEEE Transactions on Indus-trial Electronics, 34, 11-18


24. Wijesoma S.W. (1990) Robust Trajectory Following of Robots Using ComputedTorque Structure with VSS, International Journal of Control, 52, 935-962

25. Denker A., Kaynak O. (1994) Application of VSC in Motion Control Systems,in Variable Structure and Lyapunov Control, A.S.I. Zinober (Ed.), Springer-Verlag, Chapter 17, 365-383

26. Kaynak O., Harashima F., Hashimoto H. (1984) Variable Structure SystemsTheory, as applied to Sub-time Optimal Position Control with an InvariantTrajectory, Trans. IEE of Japan, Sec. E, 104, 47-52

27. Bartoszewicz A. (1988) On the Robustness of Variable Structure Systems inthe Presence of Measurement Noise, Proc. IEEE Industrial Electronics SocietyAnnual Conference, IECON’99, Aachen, Germany, 1733-1736, Aug. 31-Sept. 4

28. Slotine J.J.E., Li W. (1991) Applied Nonlinear Control, Englewood Cliffs, NJ:Prentice-Hall

29. Elmali, H., Olgac N. (1992) Robust Output Tracking Control of NonlinearMIMO Systems via Sliding Mode Technique, Automatica, 28, 145-151

30. Izosimov D. B., Utkin V. I. (1981) On Equivalence of Systems with Large Coef-ficients and Systems with Nonlinear Control, Automation and Remote Control,11, 189-191

31. Tunay I., Kaynak O. (1996) Provident Control of an Electrohydraulic Servowith Experimental Results, Mechatronics, 6, 3, 249-260

32. Tunay I., Kaynak O. (1995) A New Variable Structure Controller for AffineNonlinear Systems with Non-matching Uncertainties, International Journal ofControl, 62, 4, 917-939

33. Yen J., Langari R. (1999) Fuzzy Logic, PTR Prentice-Hall, New Jersey34. Passino K.M., Yurkovich S. (1998) Fuzzy Control, Addison-Wesley, California35. Wang L.-X. (1994) Adaptive Fuzzy Systems and Control, Design and Stability

Analysis, PTR Prentice-Hall36. Efe M.O., Kaynak O., Wilamowski B.M. (2000) Creating a Sliding Mode in a

Motion Control System by Adopting a Dynamic Defuzzification Strategy in anAdaptive Neuro Fuzzy Inference System, Proc. IEEE Int. Conf. on IndustrialElectronics, Control and Instrumentation, IECON-2000, Nagoya, Japan, 894-899, Oct. 22-28

37. Efe M.O., Kaynak O. (1999) A Comparative Study of Neural Network Struc-tures in Identification of Nonlinear Systems, Mechatronics, 9, 3, 287-300

38. Sanner R.N., Slotine J.J.E. (1992) Gaussian Networks for Direct Adaptive Con-trol, IEEE Transactions on Neural Networks, 3, 6, 837-863

39. Sira-Ramirez H., Colina-Morles E. (1995) A Sliding Mode Strategy for Adap-tive Learning in Adalines, IEEE Transactions on Circuits and Systems - I:Fundamental Theory and Applications, 42, 12, 1001-1012, December

40. Hsu L., Real J.A. (1997) Dual Mode Adaptive Control Using Gaussian NeuralNetworks, Proc. of the 36th Conference on Decision and Control, (CDC), NewOrleans, LA, 4032-4037

41. Hsu L., Real J.A. (1999) Dual Mode Adaptive Control, Proc. of the IFAC’99World Congress, Beijing, K, 333-337

42. Yu X., Zhihong M., Rahman S.M.M. (1998) Adaptive Sliding Mode Approachfor Learning in a Feedforward Neural Network, Neural Computing & Applica-tions, 7, 289-294

43. Parma G.G., Menezes B.R., Braga A.P. (1998) Sliding Mode Algorithm forTraining Multilayer Artificial Neural Networks, Electronics Letters, 34, 1, 97-98, January


44. Sira-Ramirez H., Colina-Morles E., Rivas-Echevverria F. (2000) Sliding Mode-Based Adaptive Learning in Dynamical-Filter-Weights Neurons, InternationalJournal of Control, 73, 8, 678-685

45. Efe M.O., Kaynak O., Yu X. (2000) Sliding Mode Control of a Three Degreesof Freedom Anthropoid Robot by Driving the Controller Parameters to anEquivalent Regime, Transactions of the ASME: Journal of Dynamic Systems,Measurement and Control, 122, 4, 632-640, December

46. Efe M.O., Kaynak O. (2000) On Stabilization of Gradient Based TrainingStrategies for Computationally Intelligent Systems, IEEE Transactions onFuzzy Systems, 8, 5, 564-575, October

47. Efe M.O., Kaynak O., Wilamowski B.M. (2000) Stable Training of Computa-tionally Intelligent Systems By Using Variable Structure Systems Technique,IEEE Transactions on Industrial Electronics, 47, 2, 487-496, April

48. Kaynak O., Erbatur K., Ertugrul M. (2001) The Fusion of ComputationallyIntelligent Methodologies and Sliding-Mode Control - A Survey, IEEE Trans-actions on Industrial Electronics, 48, 1, 4-17, February

49. Efe M.O. (2000) Variable Structure Systems Theory Based Training Strategiesfor Computationally Intelligent Systems, Ph.D. Dissertation, Bogazici Univer-sity

50. Yu X. Efe M.O., Kaynak O. (2001) A Backpropagation Learning Frameworkfor Feedforward Neural Networks, in Proc. of the 2001 IEEE Int. Symposiumon Circuits and Systems (ISCAS’01), III, pp. 700-702, May 6-9, Sydney, Aust

51. Zhao Y. (1996) On-line Neural Network Learning Algorithm with ExponentialConvergence Rate, Electronic Letters, 32, 15, 1381-1382, July

52. Bersini H., Gorrini V. (1997) A Simplification of the Backpropagation ThroughTime Algorithm for Optimal Neurocontroller, IEEE Transactions Neural Net-works, 8, 2, 437-441, March

53. Erbatur K., Kaynak O., Sabanovic A. (1999) A Study on Robustness Propertyof Sliding Mode Controllers: A Novel Design and Experimental Investigations,IEEE Transactions on Industrial Electronics, 46, 5, 1012-1018

54. Roy R.G., Olgac N. (1997) Robust Nonlinear Control via Moving Sliding Sur-faces - n-th Order Case, Proc. of the 36th Conference on Decision and Control,San Diego, California, U.S.A., December 943-948

55. Yilmaz C., Hurmuzlu Y. (2000) Eliminating the Reaching Phase from VariableStructure Control, Transactions of the ASME, Journal of Dynamic Systems,Measurement and Control, 122, 4, 753-757, December

56. Hwang Y.R., Tomizuka M. (1994) Fuzzy Smoothing Algorithms for VariableStucture Systems, IEEE Transactions on Fuzzy Systems, 2, 4, 277-284

57. Choi S.B., Kim M.S. (1997) New Discrete-Time Fuzzy-Sliding-Mode Controlwith Application to Smart Structures, Journal of Guidance Control and Dy-namics, 20, 5, 857-864

58. Erbatur K., Kaynak O., A. Sabanovic (1996) I. Rudas, Fuzzy Adaptive SlidingMode Control of a Direct Drive Robot, Robotics and Autonomous Systems,19, 2, 215-227

59. Chen C.S., Chen W.L. (1998) Robust Adaptive Sliding-Mode Control UsingFuzzy Modeling for an Inverted-Pendulum System, IEEE Transactions on In-dustrial Electronics, 45, 2, 297-306

60. Yu X., Man Z.H., Wu B.L. (1998) Design of Fuzzy Sliding-Mode Control Sys-tems, Fuzzy Sets and Systems, 95, 3, 295-306


61. Yoo B., Ham W. (1998) Adaptive Fuzzy Sliding Mode Control of NonlinearSystem, IEEE Transactions on Fuzzy Systems, 6, 2, 315-321

62. Ha Q.P. (1996) Robust Sliding Mode Controller with Fuzzy Tuning, ElectronicsLetters, 32, 17, 1626-1628

63. Ha Q.P. (1997) Sliding Performance Enhancement with Fuzzy Tuning, Elec-tronics Letters, 33, 16, 1421-1423

64. Ertugrul M., Kaynak O. (2000) Neuro Sliding Mode Control of Robotic Ma-nipulators, Mechatronics, 10, 1-2, 243-267


Sliding Mode Control for Systems with FastActuators: Singularly Perturbed Approach

Leonid M. Fridman

Chihuahua Institute of Technology, Av. Tecnologico 2909, Chihuahua, Chih.,31160, Mexico

Abstract. Singularly perturbed relay control systems with second order slidingmodes are considered for the modeling of sliding mode control systems with fastactuators. For sliding mode control systems with fast actuators, sufficient conditionsfor the exponential decreasing of the amplitude of chattering and unlimited growthof frequency are found. The connection between the stability of actuators and thestability of the plant on the one hand and the stability of the sliding mode system asthe whole on the other hand is investigated. The algorithm for correction of slidingmode equations is suggested for taking into account the presence of fast actuators.Algorithms are proposed to solve the problem of eigenvalues assignment or optimalstabilization for sliding motions using the additional dynamics of fast actuators.

1 Introduction

The chattering phenomenon is one of the major problems in modern slidingmode control (see for example [2], reference in [17], [18]). The presence of fastactuators is one basic reasons for chattering occurring in sliding mode controlsystems ([3],[17], and [18]). A specific feature of a sliding mode system withfast actuators is the following: a relay control is transmitted to the inputof the actuator and the continuous actuator’s output is transmitted to theinput of the plant (see fig. 1). In [2] it was shown that the behavior of sliding

Plant

Relay Controller

Actuator

Fig. 1. Control system with actuator


mode systems with fast actuators is described by singularly perturbed relaycontrol systems with higher order sliding modes and the order of sliding isthe sum of a relative degrees of the plant and the actuator (see the definitionof sliding order in the chapter of Bartolini and Levant in this book).

The main specific features of relay control systems with higher order slid-ing are the followings (see [1],[6]):- second order sliding modes could be asymptotically stable;- sliding modes of the order three and more are unstable, but a stable periodicoscillation can occur.

The chattering phenomenon for sliding mode control systems with fastactuators, whose behaviour is described by singularly perturbed relay con-trol systems with the order of sliding three and more was analyzed in [7],[8]from the view point of averaging. This chapter is devoted to analysis of thechattering phenomenon in sliding mode control systems with fast actuatorsgiven by singularly perturbed relay control systems with second order slidingmodes (SPRCSSOSM).

These specific features of SPRCSSOSM have determined the motivationsof the chapter:1. Design the mathematical tools for investigating SPRCSSOSM.The motions in SPRCSSOSM have an infinite number of switches and thetime intervals between switches tend to zero. That is why for SPRCSSOSMit is impossible to use classical methods of singular perturbations theory([11],[19]). In Section 2 the following mathematical tools for investigation ofSPRCSSOSM are developed:-sufficient conditions for the exponential decreasing of the amplitude of chat-tering and the unlimited growth of frequency are found;-the reduction principal theorem is proved in which the sufficient conditionsof the equivalence for the stability of slow motions of plants and the stabilityof original systems with an actuator are found;-it is shown that the asymptotically stable slow-motions integral manifoldof a smooth singularly perturbed system, describing the motion of originalSPRCSSOSM in the second order sliding domain, is the asymptotically sta-ble slow-motions integral manifold of the original SPRCSSOSM;-an algorithm for the asymptotical representation of a slow motions integralmanifold is suggested.2. Find the stability conditions for sliding mode control systemswith fast actuators. It is well known for smooth fast actuators systems(see, for example [11]) that the actuator’s stability and the stability of plantare not enough for the stability of system at whole. In Section 3 the analysisis made of chattering in sliding mode control systems with fast actuators. Insubsection 3.3 the connection between the stability of the actuators and thestability of the plant from the one hand and the stability of the sliding modesystem as a whole from the other hand is investigated. It is shown (subsec-tion 3.2) that the definition of the motions in sliding mode, according to the

392 L.M. Fridman

equivalent control method, corresponds to the presence of fast actuators inthe control system.3. Show that it’s possible to design chattering-free sliding modecontrol systems with fast actuators in the case when the order ofsliding in a complete model of the system is 2. For sliding mode controlsystems with fast actuators with second order sliding, the sufficient conditionsfor the exponential decreasing of the amplitude of chattering and unlimitedgrowth of frequency are found in Subsection 2.2.4. Obtain the algorithms for correction of sliding mode equationsfor taking into account the presence of fast actuators in controlsystems. It is well known (see for example [11]) that for a wide class ofcontrol systems with fast actuators the correction of slow motion is usefulin order to design a control system with better accuracy. The algorithm forcorrecting the sliding mode equations is suggested in subsection 3.1 for takinginto account the presence of fast actuators. In subsection 3.4 it is shown, thatwhenever the sliding motions of the plant are stable, but not asymptoticallystable, it is obligatory to make a correction to the sliding mode equationstaking into account the presence of fast actuators in the system.5. Suggest a control algorithm, which allows for ensuring the de-sired behavior of a system in the sliding domain, using actuatordynamics.One of the popular control design approaches for smooth systems with fastactuators is the composite control method (see [11]), which guarantees thedesired properties of slow or fast motions to be achieved. In Sections 4 and5 control algorithms are suggested for solving the problems of eigenvalue as-signment or optimal stabilization for sliding motions in systems with fastactuators.

These results leads to the following conclusion: it is possible to save oneorder of derivative to obtain a sliding mode system without chattering. Infact, one of the basic methods to avoid chattering is the design of switchingsurfaces, which include actuator variables and for having the first order slidingmode in the complete model of system (see, for example [17],[18],[14]). Butit’s not easy to measure the actuator variables and to obtain those values fromdifferentiators. As it was shown in [12], if µ is the actuator time constant, εis the noise amplitude, and k is the order of sliding for the i-th derivative, wecan have an accuracy of not more than ( ε

µ )(k−i)/k. That is why if we can save

one order of derivative, we improve the accuracy of the system by at least offactor of ( ε

µ )1/k.

For electromechanical or hydraulic systems ([18], [10]) typical actuatorsare of relative degree two. For this case the corresponding complete model ofthe sliding mode control system can be described by a singularly perturbedsystems with order of sliding three. In this case the following scenario is use-ful:- Find with a differentiator the value of the input derivative and include it

393Sliding Mode Control for Systems with Fast Actuators

into the equation of the switching surface;- Satisfy the sufficient conditions for the exponential decreasing of the chat-tering amplitude and unlimited growth of frequency for the complete modelof the system.

2 Mathematical Tools

2.1 Problem Formulation

The general model of a sliding mode control system with a fast actuator hasthe following form (see [3],[5] )

µdz/dt = f(t, z, s, x, u(s)), ds/dt = g1(t, z, s, x), dx/dt = g2(t, z, s, x), (1)

where s ∈ R, x ∈ Rn are variables describing the behaviour of the plant,z ∈ Rm is vector describing the behaviour of the actuator, u(s) = sign (s) isa relay control, f, g1, g2 are sufficiently smooth functions of their arguments,µ is the actuator time constant. The specific feature of the system (1) is thefollowing: the equations for plant’s variables s, x in (1) don’t contain the relaycontrol u(s) but it is included in equations for the fast variable z describingactuator dynamics. This means that there is no first order sliding mode inthe system (1) and only the second order sliding mode can occur.

Ignoring the dynamics of the actuator, i.e. having accepted µ = 0 andexpressing z from the equation f(z0, s, x, u(s)) = 0 according to the formulaz0 = ϕ(s, x, u(s)), we obtain the reduced system

ds/dt = g1(t, ϕ(s, x, u(s)), s, x) = F1(t, s, x, u(s)),dx/dt = g2(t, ϕ(s, x, u(s)), s, x) = F2(t, s, x, u(s)).

(2)

Here we suppose that for system (2) the sufficient conditions for existence ofa stable sliding mode

F1(t, 0, x, 1) < 0, F1(t, 0, x,−1) > 0 (3)

hold and the motion into this mode are described by the equations of equiv-alent control method (see for example [17])

dx/dt = F2(t, 0, x, ueq(t, x)), F1(t, 0, x, ueq(t, x)) = 0. (4)

This means that for the original system (1) and the reduced system (2) twoqualitatively different kinds of motion occur:- in the original system (1) there is no first order sliding mode,- however in the reduced system (2), the sufficient conditions for its existenceare held.

For the relay system (1) it’s impossible to use classical methods of singularperturbation based on separation of the spectrum in slow and fast parts, dueto the infinite number of switches (see [1], [5]). On the other hand the motion

394 L.M. Fridman

of (1) in the second order sliding mode are described by the smooth singularlyperturbed system. Under some conditions (see for example [11]) this smoothsystem has an asymptotically stable slow-motion integral manifold. In thissection we will show that this manifold is the slow-motion integral manifoldfor the original system (1) and prove the reduction principle theorem toreduce the problem of investigating stability in the system (1) to the stabilityproblem for (4).

2.2 Decomposition Theorem

System Transformation into a Convenient Form for the Analysis

We shall develop the mathematical tools for the case, when solutions of therelay control system (1) are determined uniquely. It’s true (see [11]) for a wideclass of such systems in which f is linearly depending on function U(t, x, u(s))which satisfies the inequality

U1|s| < sU(t, x, u(s)) < U2|s|, (U2 > U1 > 0)

for all (t, s, x). Let us make three substitutions of variables in system (1).1. Here we consider the case when, in system (1), there exists a stable

second order sliding mode. In such case g,1z(t, z, s, x) = 0. Suppose that zm

is the last coordinate for the vector z and g,1zm

(t, z, s, x) = 0. Then we canintroduce the variable σ = ds/dt = g1(t, z, s, x) instead of zm in the system(1).

2. It’s reasonable to consider sliding mode control systems with a stablefast actuator. From a mathematical viewpoint this means that, according tothe boundary layer method (see for example [19]) and conditions (3) ensur-ing the existence of a stable first order sliding mode, one can conclude thatthe solution of (1) starting far from switching surface s = 0 will reach theneighbourhood of the switching surface with radius O(µ) after a finite time.It allows us to examine only solutions of (1), starting in the neighbourhood ofthe switching surface with radius O(µ). That’s why it’s possible to introducethe new variable ξ = s/µ instead of variable s in (1) . After this the system(1) takes the form

µdz/dt = f1(t, z, σ, µξ, x) + d(t, x)U(t, x, u(ξ)),µdσ/dt = f2(t, z, σ, µξ, x) + b(t, x)U(t, x, u(ξ)), µdξ/dt = σ,dx/dt = g2(t, z, σ, µξ, x).

(5)

3. Let us eliminate relay control from the first equation of the system (5).In fact, after the substitution of variables η = z− d(t, x)σ/b(t, x), system (1)takes the form

µdη/dt = υ1(t, η, σ, µξ, x, µ),µdσ/dt = υ2(t, η, σ, µξ, x, µ) + b(t, x)U(t, x, u(ξ)), µdξ/dt = σ,dx/dt = υ3(t, η, σ, µξ, x).

(6)


The specific feature of system (6) is that only the second order slidingmode can occur in it, and the motion in this mode is determined by theequations

µdη/dt = υ1(t, η, 0, 0, x, µ), dx/dt = υ3(t, η, 0, 0, x, µ). (7)

System (7) has an asymptotically stable slow-motion integral manifold η =h(t, x, µ) if the following conditions are held:

I. Equation υ1(t, η, 0, 0, x, 0) = 0 has an isolated solution η = h0(t, x) atall (t, x) ∈ R×Rn .

II. Functions υi (i = 1, 3), h0 have second order continuous derivativesin the domain

Ω = (t, η, x, µ) ∈ R×Rm−1 ×Rn × (0, µ0) : |η − h0(t, x)| < δ,where δ > 0, |.| is the euclidean norm .

III. For all (t, x, µ) ∈ R ∈ Rn × (0, µ0)

Re Spec ∂υη(t, h0(t, x), 0, 0, x, µ)/∂η < −κ ≤ 0

After the substitution of variables η = η − h(t, x, µ) and expansion in theseries towards to η, σ, ξ degrees at the point (0, 0, 0), the system (6) takes theform

µdη/dt = B11(t, x, µ)η +B12(t, x, µ)σ++µB13(t, x, µ)ξ + ϕ1(t, η, σ, µξ, x, µ),µdσ/dt = B21(t, x, µ)η +B22(t, x, µ)σ + µB23(t, x, µ)ξ++ϕ2(t, η, σ, µξ, x, µ) + b(t, x)U(t, x, u(ξ)), µdξ/dt = σ,

(8)

dx/dt = ϕ3(t, η, σ, µξ, x), (9)

where ϕ1(t, η, 0, 0, x, µ) = 0 and by y = (η, σ, ξ) → 0 the following condi-tions hold ϕ2(t, η, σ, µξ, x, µ) = ϕ2(t, 0, 0, 0, x, µ)+o(|y|), ϕ1(t, η, σ, µξ, x, µ) =o(|y|) everywhere in Ω = (t, η, x, µ) ∈ R×Rm−1 ×Rn × (0, µ0) | |η| < δ.

Exponential Stability of Fast Motions

Let’s denote |y|∗ =√|η|2 + |σ|2 + |ξ|. Suppose that for all (t, x) ∈ R × Rn

conditions I-III are satisfied and, moreover,IV.B22(t, x, 0) < −α < 0, b(t, x) < −α < 0, |ϕ2(t, 0, 0, 0, x, µ)| < αU1.In appendix 1 the following lemma is proved.

Lemma 1. If conditions I-IV are true, there exist constants K1 > 0,K2 >0, γ > 0 and W some neighbourhood of the origin in the state space of vari-ables y = (ηT , σ, ξ)T , such that for all (t0, y0, x0) ∈ Ω′ = R+ ×W × Rn thefollowing inequality holds

|y(t, µ)|∗ ≤ K1|y0|∗eγ(t−t0)/µ ≤ K2e−γ(t−t0)/µ. (10)

396 L.M. Fridman

Decomposition Theorem

Consider a solution of (8),(9) only starting in Ω′. Then the x(t, µ) coordinateof the solution (8),(9) will be a solution of the initial problem

dx/dt = Φ(t, y(t, µ), x, µ), x = x0,

Φ(t, y(t, µ), x, µ) = ϕ3(t, η(t, µ), σ(t, µ), µξ(t, µ), x(t, µ)).

Let us represent x(t, µ) as x(t, µ) = x(t, µ)+πx(t, µ) such that x(t, µ), πx(t, µ)are solutions of equations

dx/dt = Φ(t, 0, x, µ), x(t0) = x0, (11)

dπx/dt = Φ(t, y(t, µ), x+ πx, µ)− Φ(t, 0, x, µ), (12)

πx(t) = π0x, x0 + π0x = x0. (13)

To define the solutions of the problems (11) - (13) it is necessary to choose(x0, π0x). The following theorem shows that (x0, π0x) can be chosen in sucha way that function πx(t, µ) exponentially decreases.

Theorem 1. Suppose that for all (t, y, x), (t, y, x) ∈ Ω′ conditions I − IVare true, inequality

|Φ(t, y, x)− Φ(t, y, x)| < M(|y − y|+ |x− x|),M = supmax(t,y,x,µ)∈Ω[|dΦ(t, y, x, µ)/dx|; |dΦ(t, y, x, µ)/dy|],

where Ω = R+ ×Rm+1 ×Rn × [0, µ0], is satisfied and

µM/γ < 1, KM/(γ − µM) < C. (14)

Then for any initial points (t0, y0, x0) ∈ Ω′ the solutions of the system (8),(9)can be represented as slow and fast parts in the form:

(y(t, µ), x(t, µ)) = (0, x(t, µ)) + (πy(t, µ), πx(t, µ)).

So x(t, µ) is the solution of equation (11) with initial conditions x(0) = x0

while x0 = x0 + O(µ). The fast part of this solution πy(t, µ), πx(t, µ) sat-isfies the inequality

µ|πy(t, µ)|+ |πx(t, µ)| < µ(C +K)e−γ(t−t0). (15)

This theorem is proved in Appendix 2.

Reduction Principle

Theorem 1 and inequality (15) yield the following reduction principle theo-rem.

Theorem 2. If under the conditions of the theorem 1 the function x(t, µ)is the solution of the system (11) then (0, 0, 0, x(t, µ)) is the solution of thesystem (8), (9) and this solution will be stable (unstable, asymptotically stable)if and only if x(t, µ) is stable (unstable, asymptotically stable).


3 Analysis of Chattering in Sliding Mode ControlSystems with Fast Actuators

3.1 Algorithm of Direct Decomposition

The results of section 2 have justified the following algorithm for correctionof equivalent control method for taking into account the presence of fastactuators in sliding mode control systems.Step 1. Design of algebraic-differential equations for description ofmotions in (1) in the second order sliding mode.

Suppose that the stable second order sliding mode exists in the system(1). Then motions in this mode are determined by equations of the equivalentcontrol method

µdz/dt = f(t, z, 0, x, ueq(t, z, x, µ)) = f(t, z, x, µ)dx/dt = g2(t, z, 0, x),

(16)

g1(t, z, 0, x) = 0, (17)

d2s/dt2(t, z, 0, x, ueq, µ) = g′1zf/µ+ g′1ξ

g1 + g′1xg2|(t,z,0,x,u,µ) = 0. (18)

Step 2. Design of differential equations for the description of mo-tions in (1) in the second order sliding mode. Let us express one ofthe vector z coordinates from equation (17). Let it be, for example, its lastcoordinate zm, and the corresponding expression has the form zm = p(t, z, x),where z ∈ Rm−1 is the vector consisting of the first (m − 1) coordinates ofthe vector z. Then system (16) may be represented in the form

µdz/dt = f(t, z, x, µ), dx/dt = g2(t, z, x), (19)

where f consists of the first (m − 1) coordinates of function f at the point(t, z, p(t, z, x), x, µ).Step 3. Design of corrected equations of the equivalent controlmethod.

System (19) is a smooth singularly perturbed system. If in such systemsthe fast variable are uniformly exponentially stable, there exists the slow-motions integral manifold in the following form: z = h(t, x, µ). Motion onthat manifold is described by equations

dx/dt = g2(t, h(t, x, µ), x), z = h(t, x, µ) (20)

According to theorem 1, the x coordinate of the solutions of (1) will differfrom the solutions of equations (20) up to the fast decreasing exponent. Inthis sense, slow motion in (19) is precisely described by equations (20), and weshall call equations (20) precise equations of the equivalent control method.

Function h(t, x, µ) could be expressed as an asymptotic series h(t, x, µ) =∑∞0 µkhk(t, x) from the equation

µ[h′t + h′

xg2(t, h(t, x, µ), x)] = f(t, h(t, x, µ), x, µ). (21)

398 L.M. Fridman

Function h0(t, x) is determined by equation

f(t, h0, x, 0) = 0. (22)

This means that for µ = 0, equation (20) coincides with the equivalent controlmethod equation (4). With µ = 0 equation (20) differs from equation (4) onlyin the terms, which correspond to the presence of fast actuators in the originalsystem (1). From theorem 2 it follows that the problems of investigating thestability for the zero solution of systems (1) and (20) are equivalent.

3.2 The Systems Containing The Relay Control Nonlinearly

Consider the control system

ds/dt = −u, dx/dt = (u2 − 1)x, x, s ∈ R, u(s) = sign (s), (23)

containing the relay control u(s) nonlinearly. There is a stable sliding mode inthe system (23). Defining solutions in the sliding domain (23) are not unique.For example, on the one hand, extension (23) of this definition of into thesliding mode according A.F. Filippov [4] takes the form dx/dt = x with anunstable zero solution. On the other hand, extension of the definition of (23)into the sliding mode according to the equivalent control method takes theform

dx/dt = −x (24)

with an asymptotically stable zero solution.Suppose that a relay control is transmitted to the plant via a fast actuator

and a complicated model of a system, taking into account the presence of afast actuator, has the form

µdz/dt = −z − u, ds/dt = z, dx/dt = (2z2 − 1)x, (25)

where z ∈ R is the actuator variable and µ is the actuator time constant. Forsystem (25) theorems 1 and 2 are true. This means that the fast variables z, sare exponentially decreasing (fig. 2, fig. 3), equation (24) of the equivalentcontrol method is approximately described by the slow motions in system(25) and the zero solution of system (25) is asymptotically stable (fig. 4).

3.3 Stability of Actuators and Absence of Chattering

In this subsection we investigate the correlation between the natural condi-tions of stability of fast actuators in sliding mode control systems and theexistence of the stable first order sliding mode for a reduced system, de-scribing the behaviour of the plant without actuator on the one hand, andsufficient conditions for exponential decreasing of fast oscillations (absence ofchattering) in the original system on the other hand.


Fig. 2. Exponential decreasing of z

Fig. 3. Exponential decreasing of s

Fig. 4. Stability of slow variable x.

400 L.M. Fridman

Consider the simplest case, when the behaviour of the plant is describedby the equation

dx/dt = Ax+Bu, x ∈ Rn, u = sign (s), s = Cx ∈ R. (26)

Suppose that the relay control u ensures the stable first order sliding modeon the switching surface s = 0, and consequently

CB < 0. (27)

Consider the case when relay control is transmitted to the plant via a fastactuator, with behaviour described by equation

µdz/dt = Dz + Fx+ bu(s), z ∈ Rm, (28)

where µ is the actuator time constant. This means that the system modeltaking into account the presence of a fast actuator has the form

µdz/dt = Dz + Fx+ bu(s), dx/dt = Ax+BKz. (29)

It is natural to suppose that:

• the actuator is stable which means that

Re SpecD < 0, (30)

• the system (28),(29) for µ = 0 turn to equation (26) and consequently

KD−1b = −1, −CBKD−1b = CB < 0. (31)

Transform the system (28),(29) to the canonic form (see section 2)

µdz1/dt = D11z1 +D12σ + F11s+ F12x1,µdσ/dt = D21z1 +D22σ + F11s+ F12x1 + du(s),ds/dt = σ, dx1/dt = B11z1 +B12σ +A33s+A34x1,z1 ∈ Rm−1, x1 ∈ Rn−1, σ ∈ R.

(32)

For system (32) the conditions of stability of the second order sliding modeare

d < 0, D22 < 0. (33)

Inequality

Re SpecD11 < 0 (34)

ensure exponential decreasing of actuator variables in the second order slidingdomain. The following proposition is obvious.

Proposition 1. When actuator is SISO system (m = 1, z ∈ R) and thecondition of stability of fast actuator (30) and conditions of existence of stablefirst order sliding mode for the reduced system (26) are held chattering in thesystem σ and s is exponentially decreasing.


But it is not true just for m = 2. Condition (31) for the system (32) meansthat

(0 1)D−1

(0d

)= d

det(D11)detD

> 0.

Now from condition (30) follows that detD > 0. Conditions (31),(30) meanthat d and det(D11) have the same sign. This means, that for m = 2, condi-tions (31),(30) do not ensure exponential decreasing of chattering.

Proposition 2. Let m = 2. If conditions (31), (30), and D11 < 0 or d <0 are held the chattering in system (29), (28) is absent.

The stability of the fast actuator (30) and of the second order sliding modein (33) still does not guarantee the absence of chattering if dim z > 1. In thiscase fast oscillations may still remain in the 2-sliding mode itself. Considerthe system

µdz1/dt = z1 + z2 + η +D1x, µdz2/dt = 2z2 + η +D2x,µdη/dt = 24z1 − 60z2 − 9η +D3x+ k sign s,ds/dt = η, dx/dt = F (z1, z2, η, s, x),

where z1, z2, η, s are scalars, k < 0. It is easy to check that the spectrum ofthe matrix is −1,−2,−3 and condition (33) hold for this system. On theother hand motions in the second order sliding mode are described by thesystem

µdz1/dt = z1 + z2 +D1x, µdz2/dt = 2z2 +D2x,dx/dt = F (z1, z2, 0, 0, x).

The fast motion in this system are unstable and the absence of chattering inthe original system cannot be guaranteed.

3.4 When correction of the equivalent control method isobligatory?

Suppose that in the sliding mode control system the behaviour of the statevector s, x ∈ R is described by equation

ds/dt = −u(s), dx1/dt = x2, dx2/dt = u(s)− x1, u(s) = −sign (s). (35)

There exists a stable first order sliding mode for system (35). The motion insliding mode are described by equations

dx1/dt = x2, dx2/dt = −x1. (36)

It’s obvious that the solutions of this system are stable but not asymptoticallystable. Suppose that the relay control u(s) is transmitted to the plant with

402 L.M. Fridman

Fig. 5. Unstability of slow variables on x20x1 plain for a = −0.5

the help of a fast actuator, whose behaviour is described by variables z1, z2.The complete mathematical model of control system has the form

µdz1/dt = −z1 − x1, µdz2/dt = −z2 − sign (s),ds/dt = z2, dx1/dt = x2, dx2/dt = (a+ 1)z1 − z2 + ax1.

(37)

It’s easy to see that for system (37) theorems 1 and 2 are true and slowmotion for (37) with precision level o(µ) is described by equations (36). Onthe other hand, motion in the second order sliding mode for system (37)

µdz1/dt = −z1 − x1, dx1/dt = x2, dx2/dt = (a+ 1)z1 + ax1. (38)

Then the slow-motion manifold of systems (37) and (38) takes the form z1 =p1(µ)x1 + p2(µ)x2, where

pij(µ) = pi0 + pi1µ+ ...+ pikµk + ..., i = 1, 2.

The functions pij(µ) can be found from equation

µ(p1 p2)((

0a+ 1

)(p1 p2) +

(0 1a 0

))= −(p1 p2)− (1, 0), (39)

and consequently (p10 p20) = (−1, 0), (p11 p21) = (0, 1). This means that theslow motion in system (37) is described by equations

dx1/dt = x2, dx2/dt = −x1 + µ(a+ 1)x2 +O(µ2). (40)

From theorems 1 and 2 it follows that variables s and ds/dt = z2 are asymp-totically decreasing, but for a > −1 the zero solution of the system (37) isunstable (fig. 5) and for the a < −1 this solution is asymptotically stable(fig. 6).

This means that in the case, when the spectrum of sliding mode equationsis critical, the presence of fast actuators can change the behaviour of a systemfrom stability to instability or asymptotic stability. One can conclude thatfor the investigation of stability in the critical case, the correction of slidingmode equations is obligatory.


Fig. 6. Stability of slow variables on x20x1 plain for a = −2.

4 Additional Dynamics of Actuators in the Problem ofEigenvalue Assignment

4.1 Problem Statement

Consider the control system with the state vector (s, x) (s ∈ R, x ∈ Rn)

ds/dt = A1s+A2x+ b1u(s), dx/dt = A3s+A4x+ b2u(s), (41)

where s is the output of (41). Suppose that the control goal is for the outputto go to zero. One of the simplest and most robust methods to reach thisgoal is to design the relay control in the form u(s) = sign(s) and ensure thestable first order sliding mode on the surface s = 0. In this case the equationfor equivalent control has the form ds/dt = A2x+ b1ueq = 0, and for b1 = 0ueq = −b−1

1 A2x. Then the sliding motions in system (41) are described byequations

dx/dt = (A4 − b2b−11 A2)x. (42)

In [17] two methods to solve the design problem were proposed for the desiredequations of sliding mode:- to extend the state space by using additional dynamics and to solve theproblem of eigenvalue assignment in the extended state space;- to include the derivatives of the variable s into the equation of the switchingsurface.

In [13], [14] the fast variable describing the behaviour of the fast actuatorin the equation for the switching surface was introduced for motion controlin singularly perturbed discontinuous control systems. This approach ensuresthe existence of a first order sliding mode in the overall system. For suchsystems the composite control method ( see [11]) was used [9], [16]. Theseapproaches needed the measuring of fast variables, which is hard in realsystems.

In this section control algorithms are designed to ensure:

404 L.M. Fridman

• the absence of chattering;• the desired behaviour of system going into the sliding domain via the

dynamics of fast actuators.

In the proposed approach only the slow coordinates of the state-vector areused for control design. Moreover, the design problems are solved in thespace of sliding mode equations. On the other hand, the proposed algorithmis useful only in the case when the actuator is a MIMO system.

In [7] the problem of eigenvalue assignment is solved with the help of asimilar approach for sliding mode systems with the fast actuators describedby singularly perturbed relay control systems with the order of sliding threeor more. But, in such systems fast periodic oscillations occur. That’s why theaveraging technique was used and the problem of eigenvalue assignment wassolved with precision level O(µ).

Let us suppose that the complete model of a control system, taking intoaccount the presence of fast actuator, has the form

µdz/dt = B1z +B2s+B3x+ dvds/dt = B4z +B5s+B6x, dx/dt = B7z +B8s+B9x

(43)

where z ∈ Rm, v ∈ Rl, µ is the actuator time constant. Now we suppose thatthe conditions

rank(B4

B7

)≥ 2, rank d ≥ 2, m ≥ l ≥ 2 (44)

are held. Conditions (44) mean that the relay control is transmitted to theplant through an actuator, which is a MIMO system itself.

Ignoring actuator dynamics, having accepted µ = 0 and expressing zaccording to the formulae z0 = −B−1

1 (B2s+B3x+ dv), we obtain

ds/dt = (B5 −B4B−11 B2)s+ (B6 −B4B

−11 B3)x−B4B

−11 dv,

dx/dt = (B8 −B7B−11 B2)s+ (B9 −B7B

−11 B3)x−B7B

−11 dv.

(45)

Let us suppose that in the case, when the control law has been designed inform v = Ku(s) (K is constant vector), systems (45) and (41) coincide.

The proposed algorithm uses the singular correction of the equivalentcontrol method. Now we propose to use the control law in the form

v = Ku(s) + w. (46)

The motion on the slow-motion manifold in (43) is described by equation

dx/dt = (A4 − b2b−11 A2)x− [B7 − b2b

−11 B4]B−1

1 dw. (47)

If condition:D.1. The pair (A4 − b2b

−11 A2); (B7 − b2b

−11 B4)B−1

1 d is controllableis true, choosing the control vector for (47) in form w = Lx, we can solve theeigenvalue assignment problem for (47). This means, that we have proposed


an algorithm for the design of the control vector w, which allows us to solvethe eigenvalue assignment problem for (47). System (47) approximately de-scribes the slow motions in a small neighborhood of the switching surface ofthe system (43).

To have sufficient conditions for absence of chattering and to obtain thehigher approximations, we transform system (43) into the canonical form.Suppose that the control law (46) in the control system (43) has been de-signed. Let us make three variable changes in system (43) according to Section2.1. Consider only solutions of (43) with initial conditions s(0, µ) = O(µ) andintroduce variable ξ = s/µ in system (43).2. Introduce variable σ = dξ/dt instead one of vector z coordinates.3. Eliminate the relay control u(s) from the system equations except theequation for σ.

Then we obtain system (43) in the canonical form

µdη/dt = F11η + F12σ + µF13ξ + F14x+ d1w,µdσ/dt = F21η + F22σ + µF23ξ + F24x+ d2(Ku(s) + w),µdξ/dt = σ, dx/dt = F41η + F42σ + µF43ξ + F44x,

(48)

where Fij could only depend on µ . The motion in the second order slidingmode in (48) is described by equations

µdη/dt = F11η + F14x+ d1w, dx/dt = F41η + F44x. (49)

With µ = 0 we have F41F−111 F14 + F41 = A4 − b2b

−11 A2, −F41F

−111 d1 =

(B7 − b2b−11 B4)B−1

1 d and following theorem is true.

Theorem 3. Suppose that condition D.1 is held and for all µ ∈ [0, µ0] con-ditionsD.2. Re Spec F11 < 0,D.3. F24 < 0, d < 0hold. Then the slow motion in system (43), within the accuracy of O(µ), is de-scribed by equation (47) and there exists matrix L0, which provides for the de-sired characteristic polygon of matrix A4−b2b

−11 A2−(B7−b2b

−11 B4)B−1

1 dL0.

In the case, when more than O(µ) precision level is needed, the followingalgorithm can be used. Suppose that

Fij = Fij0 + µFij1 + µ2Fij2 + . . . .

The behaviour of slow motion in (43) with a precision level up to the fastdecreasing exponent is described by equations

dx/dt = F41H(µ) + F44x. (50)

Function H(µ) can be found in the form H(µ) = H0 +µH1 +µ2H2 + ... from

µH(F41H + F44) = F11H + F14 + d1L. (51)

406 L.M. Fridman

Let us find matrix L(µ) = L0 +µL1 +µ2L2 + ..., such that the characteristicpolynomial of matrix (F41H(µ)+F44) in (50) has the form λn+a1(µ)λn−1+a2(µ)λn−2...+ an(µ), where ai = ai0 + µai1 + µ2ai2 + ..., i = 1, ..., n.

Then H0 = −F−1110(F140 + d1L0) and consequently the matrix L can be

designed such as in theorem 3. In this case

H1 = −F−1110[F141 + d1L1 + F111H0 −H0(F410H0 + F440)]

and matrix L1 is designed as the matrix, ensuring that characteristic poly-nomial of matrix

−F410F−1110[F141 + F111H0 −H0(F410H0 + F440)]+

+F411H0 + F441 − F410F−1110d1L1

has the form λn + a11(µ)λn−1 + a12(µ)λn−2 + ...+ a1n(µ). This means thatLi can be found under the conditionD.1.1. Matrices

−F410F−1110[F141 + F111H0 −H0(F410H0 + F440)] + F411H0 + F441

and (B7 − b2b−11 B4)B−1

1 d are controllable.Let us suppose that matrices Lk(k = 0, ..., i−1) have been found. Then we

can linearly express Hi through Li from (51). Substitution of this expressioninto (51) yields

dx/dt = [A4 − b2b−11 A2 − (B7 − b2b

−11 B4)B−1

1 dL0 + ...

+µi(Φi(L0, .., Li−1)− (B7 − b2b−11 B4)B−1

1 dLi)]x,

where the Φi is the matrix depends on L0, ..., Li−1, and, consequently, ifmatrices Φi and (B7 − b2b

−11 B4)B−1

1 d are controllable, then we can designthe matrix Li such that characteristic polynomial of matrix Φi(L0, .., Li−1)−(B7−b2b

−11 B4)B−1

1 dLi has the form λn+ai1(µ)λn−1+ai2(µ)λn−2+...+ani(µ).Thus the following theorem is true

Theorem 4. Suppose that conditions of theorem 3 are held and matricesΦk and (B7 − b2b

−11 B4)B−1

1 d (k = 0, ..., i) are controllable. Then there existmatrices Lk (k = 0, ..., i) ensuring the design of a control law in the formv = Ku(s) + (L0 + µL1 + µ2L2 + ... + µiLi + . . .)x, such that the spectrumof the matrix in equation (49) and roots of equation (51) differ by an errorO(µi+1).


4.2 Example

Let us suppose that the state vector of the control system is described withequations

ds/dt = −u(s), dx1/dt = x2, dx2/dt = u(s)− x1, (52)

where s, x1, x2 ∈ R and the relay control u(s) = −sign (s) has been designed.The motion in the sliding mode in (52) are described by equations

dx1/dt = x2, dx2/dt = −x1. (53)

The spectrum of the matrix in (53) is situated on the imaginary axis. Letus suppose that the relay control u(s) is transmitted to the plant with thehelp of actuators, whose behaviour is described by variables z1, z2. Then thecomplete model of the system has the form

µdz1/dt = −z1 + v1 − x1, µdz2/dt = −z1 + v2,ds/dt = z2, dx1/dt = x2, dx2/dt = z1.

(54)

It can easily be seen that in this case, when we suppose that v1 = v2 =−sign(s), the slow motion in it is described by system (53) with an error ofO(µ). Now we shall find the control law in the form v = −sign(s)+l1x1+l2x2.In such case the system (54) can be rewritten in the form

µdz1/dt = −z1 − sign(s) + (l1 − 1)x1 + l2x2,µdz2/dt = −z2 − sign(s),ds/dt = z2, dx1/dt = x2, dx2/dt = z1.

(55)

Then the equations describing the system (55) motion in the second ordersliding mode have the form

µdz1/dt = −z1 + (l1 − 1)x1 + l2x2, dx1/dt = x2, dx2/dt = z1. (56)

System (56) has the slow motion integral manifold z = h1x1 + h2x2. Thenthe equation for h1, h2 takes the form

µ(h1 h2)((

0 1)(h1 h2) +

(0 10 0

))= −(h1 h2) + (l1 − 1 l2) (57)

and slow motion in (54) is described by equations

dx1/dt = x2, dx2/dt = h1x1 + h2x2. (58)

Let’s find such l1(µ), l2(µ), h1(µ), h2(µ) as the asymptotic series

l1(µ) = l11 + µl22 + ..., l2(µ) = l21 + µl22 + ...,

h1(µ) = h11 + µh12 + ..., h2(µ) = h21 + µh22 + ...

408 L.M. Fridman

Fig. 7. Desired behavior of slow variables on x2x1 plain

such that characteristic polynomial of the system (58) takes the form λ2 +α(µ)λ+ β(µ), where

α(µ) = α0 + µα1 + ...; β(µ) = β0 + µβ1 + ....

Substitution of the asymptotic series into (57)

(h10, h20) = (l10 − 1, l20)

(h11, h21) = (−(l10 − 1)l20 + l11,−(l10 − 1)− l220 + l21).

This means that choosing

l10 = 1− β0, l20 = −α0

l11 = −α1 + (l10 − 1)l20, l21 = −β1 + (l10 − 1) + l220

we can provide that the spectrum of the matrix in equations (58) coincideswith the roots of the desired polynomial with the accuracy of O(µ2). Desiredexponential decreasing of variables x1, x2 is shown in the figure 7.

5 Optimal Stabilization of Motions into the SlidingMode

5.1 Problem Statement

Let us suppose that the state vector (s, x) in a control system is describedby equations

ds/dt = A1s+A2x+ b1u(s), dx/dt = A3s+A4x+ b2u(s), (59)

and u(s) is the relay control, which provides the existence of a stable first or-der sliding mode. Moreover, we will suppose that the behaviour of the controlsystem, taking into account the presence of a fast actuator, is described bysystem (43); that conditions (44) are true; and we try to design the controlin the form (46) to reach two goals:


• Provide fast exponentially decreasing of chattering to s = 0 in (43);• Find an approximate solution to the optimal stabilization problem for

functional∫ ∞

0

(x∗(t)Qx(t) + w∗(t)w(t))dt, (60)

where Q is symmetrical matrix, for system trajectories of (59).

5.2 Main Theorem

Under conditions D.2,D.3 the difference between systems (43) and (49) isno more than in a fast decreasing exponent. This means that the min. offunctional (60) for systems (43) and (49) are the same with a precision levelof O(µ).

Let us follow the algorithm of the optimal stabilization problem (49),(60)provided in [15]. We will try to find the control law in the form

w = (d∗1 0)(K1 K2

K∗2 K3

)(ηx

)

where K1,K2,K3 are solutions of the following system

−K3F44 − F ∗44K3 −K2F14 − F14K

∗2 +K2d1d

∗1K2 = Q,

−K1F11 − F ∗11K1 +K2d1d

∗1K1 − µ(K∗

2F41 + F ∗41K2) = 0,

−K3F41 −K2F11 − F ∗14K1 +K2d1d

∗1K1 − µF ∗

44K2 = 0.(61)

Having accepted µ = 0 in (61), we can obtain K1 = 0, K2 = −K3F41F−111 .

This means that K3 is a solution of Riccatti equation

−K3(F44 − F41F−111 F14)− (F44 − F41F

−111 F14)∗K3+

+K3(F41F−111 d1)(F41F

−111 d1)∗K3 = Q.

AssumeD.4. The pair q, F44 − F41F

−111 d1, where q∗q = Q, is observable.

If the condition D.4 holds the unique solution of equation for K exists andthe following theorem is true.

Theorem 5. Let us suppose that for system (41) for all µ ∈ [0, µ0] the con-ditions D.1-D.4 are held. Then we can find the solution of the problem ofoptimal stabilization (43),(60) in the form

w =(d∗1 0

) (0 −K3F41F

−111

(−K3F41F−111 )∗ K3

)(ηx

)

and consequently this solution and solution of the problem (47),(60) are thesame with precision level up to O(µ).

410 L.M. Fridman

5.3 Example

Let us suppose that the state vector of a control system is described by theequations

ds/dt = −u(s), dx1/dt = x2, dx2/dt = u(s)− x1, s, x1, x2 ∈ R (62)

and the relay control u(s) = −sign (s) has been designed. The motion in thesliding mode in (62) are described by equations

dx1/dt = x2, dx2/dt = −x1. (63)

The spectrum of matrix in (63) is imaginary. Let us suppose that the relaycontrol u(s) is transmitted to the plant with the help of actuators whosebehaviour is described by variables z1, z2 and the complete model of systemhas the following form

µdz1/dt = −z1 + v1 − x1, µdz2/dt = −z1 + v2,ds/dt = z2, dx1/dt = x2, dx2/dt = z1.

(64)

It can be easily seen that in case when we suppose that v1 = v2 = −sign(s),system (64) takes the form

llµdz1/dt = −z1 − sign (s)− x1, µdz2/dt = −z2 − sign (s),ds/dt = z2, dx1 = x2, dx2/dt = z1.

The slow motion in it, with an error of O(µ) is described by system (63).Let us try to reach two goals:

a) Provide the fast exponentially decreasing of chattering in the system (64)to the surface s = 0;b) Find an approximate solution of the optimal stabilization problem forfunctional∫ ∞

0

(3(x21(t) + x2

2(t)) + w2(t))dt. (65)

To solve this problem with high precision, given the control law in the formv1 = −sign (s) + w, v2 = −sign (s) it is necessary to obtain the minimum offunctional (65) on the trajectories of the system

dx1/dt = x2, dx2/dt = −x1 + w. (66)

We shall find vector w in the form

w = (0 1)(K11 K12

K∗12 K22

)(x1

x2

).

Then

2K12 +K212 = 3, K11 −K22 −K12K22 = 0, 2K12 −K2

22 = −3.

This means that choosing the control law in the form

w = (0 1)(2√5 1

1√5

) (x1

x2

)

we have the solution of the problem (62), (65) with an error of O(µ).


6 Conclusion

1. The mathematical tools for investigation of SPRCSSOSM are developed.2. The sufficient conditions under which the oscillations in the sliding modecontrol systems, with fast actuators, whose behaviour are described by SPRC-SSOSM, have the following structure:- the oscillations develop in the direction orthogonal to the switching surfacedesigned according to measuring results. An amplitude of these oscillationsexponentially decreases, time intervals between switches vanish and the fre-quency of such oscillations is infinitely increasing;- the oscillations in the second order sliding mode which are described by asmooth singularly perturbed system of differential equations, and the slow-motion integral manifold of this system is the stable slow-motion integralmanifold of the original system.Due to this fact it was shown that it’s possible to design chattering-free slid-ing mode control systems with fast actuators in the case when the order ofsliding in a complete model is 2.3. It is shown that it is possible to save one order of derivative to obtain thesliding mode system without chattering.4. It is proved that in the general case, when the plant contains the relaycontrol nonlinearly, the equations of the equivalent control method for thesliding motions of the plant are approximately describing the slow motion inthe original SPRCSSOSM and correspond to the presence of fast actuatorsin a sliding mode control system.5. The connection between the stability of the actuators and the stability ofthe plant on the one hand and the stability of the sliding mode system as awhole on the other hand is investigated.6. The algorithm for the correction of the sliding mode equation is proposed.In the case, when the linear part of the sliding mode equations have a criticalspectrum, it is obligatory to correct the equations of the sliding motion in or-der to take into account the presence of fast actuators in the system, becausethe presence of such devices may cause change to the system behaviour fromstability to asymptotic stability or instability.7. The algorithms are proposed allowing us to solve problems of eigenvalueassignment or optimal stabilization for sliding motion using the additionaldynamics of fast actuators.

Appendix 1. Exponential Stability of Fast Motions

For investigating stability of fast motion in (8),(9) it is necessary to introducein (8) new fast time τ = t/µ and rewrite system (8) in form

dη/dτ = B11(t, x, µ)η +B12(t, x, µ)σ++µB13(t, x, µ)ξ + ϕ1(t, η, σ, µξ, x, µ),dσ/dτ = B21(t, x, µ)η +B22(t, x, µ)σ + µB23(t, x, µ)ξ++ϕ2(t, η, σ, µξ, x, µ) + b(t, x)U(t, x, u(ξ)), dξ/dτ = σ,

(67)

412 L.M. Fridman

where (t, x) are parameters. To analyze the stability of relay systems (67) letus examine the Lyapunov function

E = η∗Sη + σ2 − ξ[2b(t, x)U(t, x, u(ξ))+

+2ϕ2(t, η, σ, µξ, x, µ) +B22σ + 2η∗SB12 + 2B21η].

Here S(t, x, µ) is positive definite solution of equation SB11+B∗11S = −Im−1.

Then there are some constants κ2 > κ1 > 0 such that inequality

κ1|η|2 ≤ η∗Sη ≤ κ2|η|2 (68)

is true uniformly for (t, x, µ) ∈ R+ ×Rn × [0, µ0]. Taking into account con-dition (iv) and inequality (68) one can conclude the following estimation forthe Lyapunov function

κ3|y|2∗ ≤ E(t, y, x, µ) ≤ κ4|y|2∗ (κ4 > κ3 > 0). (69)

which is true for (t, y, x, µ) ∈ R+ × Rn × U1 × (0, µ0), where U1 is someneighbourhood of zero in the state space of variables y = (η, σ, ξ). In thiscase

dE/dτ = −|η|2 +B22σ2 − ξB22[bU(t, x, u(y), 0)+

+ϕ2(t, 0, 0, 0, x, 0)] + ξN1 + ξN2 +N3,

where functions Bij , bj are computed at the point (t, x, 0). For N1, N2, N3

at y → 0, µ → 0 uniformly on (t, x, µ) ∈ R+ × Rn × [0, µ0] the followingasymptotic representations are true

N1(t, y, x, µ) = o(1), N2(t, y, x, µ) = O(µ), N3(t, y, x, µ) = o(|y|2).

Taking into account conditions (iii)− (iv) we have the inequality

κ5|y|2∗ ≤ −dE/dτ ≤ κ6|y|2∗ (κ6 > κ5 > 0), (70)

which is true at (t, y, x, µ) ∈ R+ ×Rn × U2 × [0, µ0], where U2 some neigh-bourhood of the origin in the state space of y.

From (69) and (70) one can conclude that there are some constants κ8 >κ7 > 0, in the neighbourhood of the origin W = U1 ∩ U2 such that for any(t, y, x, µ) ∈ R+ ×W ×Rn × [0, µ0] the following inequality

κ7E ≤ −dE/dτ ≤ κ8E. (71)

is true. Lemma 1 follows from inequality (71).


Appendix 2. Proof of Decomposition Theorem

Consider the system (11),(12). Let’s design an integral manifold of the system(11),(12) in form S = (t, x, πx) ∈ R+ ×Rn ×Rn : πx = H(t, x, µ), wherethe function H(t, x, µ) is continuous on R+ ×Rn × [0, µ0] and the followinginequality is true

sup|exp(γt/µ)H(t, x, µ)| < µd, (t, x) ∈ R×Rn, (72)

The constant d > 0 in (72) will be defined later. Denote as U the metricalspace of continuous functions R+×Rn× [0, µ0] → Rn, satisfying the inequal-ity (72) with the metric ρ(H, H) = sup|exp(γt/µ)(H(t, x, µ)−H(t, x, µ))|, for(t, x, µ) ∈ R+ ×Rn × [0, µ0]. The space U is complete metric space.

In this case function πx = H(t, x, µ) ∈ U is the solution of the equation

H = P(H), (73)

where

P(H)(τ, ξ, µ) = −∫ ∞

t

[Φ(θ, y(θ, µ), φ(θ, µ) +H(θ, φ(θ, µ), µ), µ)−

−Φ(θ, 0, φ(θ, µ), µ)]dθ,

and φ(θ, µ) is the solution of Cauchy problem dφ/dθ = Φ(θ, 0, φ, µ), φ(t) = ξ.Let’s show that operator P from (73) transforms U into itself. Taking into

account (72) and (73) one can conclude that

|exp(γt/µ)P(H)(t, ξ, µ)| < M exp(γt/µ)∫ ∞

τ

[|H(θ, φ(θ, µ), µ)|+

+|y(θ, µ)|]dθ <M

γ[µd+ C|y0|∗].

Now it’s possible to choose such d that for any y0 ∈ W ( W from lemma1) the inequality M

γ [µd + C|y0|∗] ≤ d is true. This means that operator Ptransforms the space U into itself. Similarly,

exp(γt/µ)|P(H)(t, ξ, µ)− P(H)(τ, ξ, µ)| ≤

≤ exp(γt/µ)∫ ∞

t

|Φ(θ, φ(θ, µ) +H, y(θ, µ), µ)− Φ(θ, φ(θ, µ)+

+H, y(θ, µ), µ)|dθ ≤ exp(γt/µ)∫ ∞

t

M |H − H|dθ ≤ µM

γρ(H, H),

which means that operator P is a contraction operator on U . Then, theoperator P has the unique fixed point corresponding to the function πx =H(t, x, µ). Moreover, from (72) one can conclude that for all (t, x, µ) ∈ R+ ×Rn × [0, µ0] the inequality |H(t, x, µ))| < µd exp(γt/µ) is true.

414 L.M. Fridman

References

1. Anosov D.V. (1959) On stability of equilibrium points of relay systems. Auto-matica i telemechanica ( Automation and Remote Control) 10, 135–149 (Rus-sian)

2. Bartolini G. (1990) Chattering phenomena in discontinuous control systems.Int. Journ. of System Science, 20, 2471 - 2481

3. Bondarev A.G., Bondarev S.A. et al. (1985) On stability of equilibrium points ofrelay systems. Automatica i telemechanika ( Automation and Remote Control)46, 679–684

4. Filippov A.F. (1988) Differential Equations with Discontinuous Right HandSide. Kluwer Publishers, Dodrecht

5. Fridman L., Levant A. (1996) Higher order sliding modes as the natural phe-nomenon in control theory. In: F. Garafalo and L. Glielmo. (Eds.) Robust Con-trol via Variable Structure and Lyapunov Techniques, Lecture notes in computerand information science, 217, Springer - Verlag, Berlin , 107-133

6. Fridman L., Levant A. (2001) Higher order sliding modes. in: J.P Barbot,W.Perruguetti (Eds.) Sliding mode control in engineering, Marcel Dekker: NewYork, 51-100.

7. Fridman L.M. (1999) The problem of chattering: an averaging approach. In: Young K.K.D., Ozguner U. (Eds.) Variable Structure Systems, Sliding Modeand Nonlinear Control, Lecture Notes in Control and Information Sciences, 247,Springer - Verlag, Berlin, 363-386

8. Fridman L.M. (2001) An averaging approach to chattering. IEEE Transactionof Automatic Control, 46(8), 1260-1265

9. Heck B.S. (1991) Sliding mode control for singularly perturbed system. Int.Journ. of Control, 53, 985 - 1001

10. Krupp D., Shkolnikov I.A., Y.B. Shtessel (2000) 2- sliding mode control for non-linear plants with parametric and dynamic. Proc. of AIAA Guidance Navigationand Control Conference, Denver,CO, August, 2000, AIAA paper 2000-3965

11. Kokotovic P.V., Khalil H.K. et al. (1986) Singular Perturbation Methods inControl: Analysis and Design. Academic Press, London

12. Levant A. (1998) Robust exact differentiation via sliding mode technique. Au-tomatica, 34, 379-384

13. Sira-Ramires H. (1988) Sliding regimes on slow manifolds of systems with fastactuators. Int. J. of System Science, 37, 875-887

14. Sira-Ramires H. (1989) Sliding regimes in general non - linear systems: a relativedegree approach. Int. J. of Control. 50, 1487-1506

15. Sobolev V.A. (1992) Singular perturbation in linear-quadratic problem of op-timal control. Automation and Remote Control. 53, 53-63

16. Su W.-C. (1999) Sliding mode control for singularly perturbed systems. Int.J. of Control. 72, 990-995

17. Utkin V.I. (1992) Sliding Modes in Control Optimization. Springer Verlag,Berlin

18. Utkin V., Guldner J. et al. (1999) Sliding Modes in Electromechanical Systems.Taylor and Francis, London

19. Vasilieva A.B., Butusov V.F. et al. (1995) The Boundary Layer Method forSingular Perturbation Problems. SIAM, Philadelphiia
